Power expressions (expressions with powers) and their transformation. Lesson "Multiplication and division of powers" Subtraction of numbers with the same powers

It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds equal powers of identical variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is equal to 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3.

It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Multiplying powers

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.

Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n;

And a m is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents of the powers.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y -n .y -m = y -n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.

So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.

Division of degrees

Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.

Thus, a 3 b 2 divided by b 2 is equal to a 3.

Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$

Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing degrees with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1. That is, $\frac(yyy)(yy) = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.

Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3

The rule is also true for numbers with negative values of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$

It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce the exponents by $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Decrease the exponents by $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.

3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.

Lesson content

What is a degree?

Degree called a product of several identical factors. For example:

2 × 2 × 2

The value of this expression is 8

2 × 2 × 2 = 8

The left side of this equality can be made shorter - first write down the repeating factor and indicate above it how many times it is repeated. The repeating multiplier in this case is 2. It is repeated three times. Therefore, we write a three above the two:

2 3 = 8

This expression reads like this: “ two to the third power equals eight" or " The third power of 2 is 8."

The short form of notation for multiplying identical factors is used more often. Therefore, we must remember that if another number is written above a number, then this is a multiplication of several identical factors.

For example, if the expression 5 3 is given, then it should be borne in mind that this expression is equivalent to writing 5 × 5 × 5.

The number that repeats is called degree basis. In the expression 5 3 the base of the power is the number 5.

And the number that is written above the number 5 is called exponent. In the expression 5 3, the exponent is the number 3. The exponent shows how many times the base of the exponent is repeated. In our case, base 5 is repeated three times

The operation of multiplying identical factors is called by exponentiation.

For example, if you need to find the product of four identical factors, each of which is equal to 2, then they say that the number is 2 raised to the fourth power:

We see that the number 2 to the fourth power is the number 16.

Note that in this lesson we are looking at degrees with natural exponent. This is a type of degree whose exponent is a natural number. Recall that natural numbers are integers that are greater than zero. For example, 1, 2, 3 and so on.

In general, the definition of a degree with a natural exponent looks like this:

Degree of a with natural indicator n is an expression of the form a n, which is equal to the product n factors, each of which is equal a

Examples:

You should be careful when raising a number to a power. Often, through inattention, a person multiplies the base of the exponent by the exponent.

For example, the number 5 to the second power is the product of two factors, each of which is equal to 5. This product is equal to 25

Now imagine that we inadvertently multiplied base 5 by exponent 2

There was an error because the number 5 to the second power is not equal to 10.

Additionally, it should be mentioned that the power of a number with exponent 1 is the number itself:

For example, the number 5 to the first power is the number 5 itself

Accordingly, if a number does not have an indicator, then we must assume that the indicator is equal to one.

For example, the numbers 1, 2, 3 are given without an exponent, so their exponents will be equal to one. Each of these numbers can be written with exponent 1

And if you raise 0 to some power, you get 0. Indeed, no matter how many times you multiply anything by itself, you get nothing. Examples:

And the expression 0 0 makes no sense. But in some branches of mathematics, in particular analysis and set theory, the expression 0 0 may make sense.

For practice, let's solve a few examples of raising numbers to powers.

Example 1. Raise the number 3 to the second power.

The number 3 to the second power is the product of two factors, each of which is equal to 3

3 2 = 3 × 3 = 9

Example 2. Raise the number 2 to the fourth power.

The number 2 to the fourth power is the product of four factors, each of which is equal to 2

2 4 =2 × 2 × 2 × 2 = 16

Example 3. Raise the number 2 to the third power.

The number 2 to the third power is the product of three factors, each of which is equal to 2

2 3 =2 × 2 × 2 = 8

Raising the number 10 to the power

To raise the number 10 to a power, it is enough to add after one a number of zeros equal to the exponent.

For example, let's raise the number 10 to the second power. First, we write down the number 10 itself and indicate the number 2 as an indicator

10 2

Now we put an equal sign, write one and after this one we write two zeros, since the number of zeros must be equal to the exponent

10 2 = 100

This means that the number 10 to the second power is the number 100. This is due to the fact that the number 10 to the second power is the product of two factors, each of which is equal to 10

10 2 = 10 × 10 = 100

Example 2. Let's raise the number 10 to the third power.

In this case, there will be three zeros after the one:

10 3 = 1000

Example 3. Let's raise the number 10 to the fourth power.

In this case, there will be four zeros after the one:

10 4 = 10000

Example 4. Let's raise the number 10 to the first power.

In this case, there will be one zero after the one:

10 1 = 10

Representation of numbers 10, 100, 1000 as powers with base 10

To represent the numbers 10, 100, 1000 and 10000 as a power with a base of 10, you need to write down the base 10, and as an exponent specify a number equal to the number of zeros of the original number.

Let's imagine the number 10 as a power with a base of 10. We see that it has one zero. This means that the number 10 as a power with a base of 10 will be represented as 10 1

10 = 10 1

Example 2. Let's imagine the number 100 as a power with a base of 10. We see that the number 100 contains two zeros. This means that the number 100 as a power with a base of 10 will be represented as 10 2

100 = 10 2

Example 3. Let's represent the number 1,000 as a power with a base of 10.

1 000 = 10 3

Example 4. Let's represent the number 10,000 as a power with a base of 10.

10 000 = 10 4

Raising a negative number to the power

When raising a negative number to a power, it must be enclosed in parentheses.

For example, let's raise the negative number −2 to the second power. The number −2 to the second power is the product of two factors, each of which is equal to (−2)

(−2) 2 = (−2) × (−2) = 4

If we did not enclose the number −2 in brackets, it would turn out that we are calculating the expression −2 2, which not equal 4 . The expression −2² will be equal to −4. To understand why, let's touch on some points.

When we put a minus in front of a positive number, we thereby perform operation of taking the opposite value.

Let's say you're given the number 2, and you need to find its opposite number. We know that the opposite of 2 is −2. In other words, to find the opposite number for 2, just put a minus in front of this number. Inserting a minus before a number is already considered a full-fledged operation in mathematics. This operation, as stated above, is called the operation of taking the opposite value.

In the case of the expression −2 2, two operations occur: the operation of taking the opposite value and raising it to a power. Raising to a power has a higher priority than taking the opposite value.

Therefore, the expression −2 2 is calculated in two stages. First, the exponentiation operation is performed. In this case, the positive number 2 was raised to the second power

Then the opposite value was taken. This opposite value was found for the value 4. And the opposite value for 4 is −4

−2 2 = −4

Parentheses have the highest execution priority. Therefore, in the case of calculating the expression (−2) 2, the opposite value is first taken, and then the negative number −2 is raised to the second power. The result is a positive answer of 4, since the product of negative numbers is a positive number.

Example 2. Raise the number −2 to the third power.

The number −2 to the third power is the product of three factors, each of which is equal to (−2)

(−2) 3 = (−2) × (−2) × (−2) = −8

Example 3. Raise the number −2 to the fourth power.

The number −2 to the fourth power is the product of four factors, each of which is equal to (−2)

(−2) 4 = (−2) × (−2) × (−2) × (−2) = 16

It is easy to see that when raising a negative number to a power, you can get either a positive answer or a negative one. The sign of the answer depends on the index of the original degree.

If the exponent is even, then the answer will be positive. If the exponent is odd, the answer will be negative. Let's show this using the example of the number −3

In the first and third cases the indicator was odd number, so the answer became negative.

In the second and fourth cases the indicator was even number, so the answer became positive.

Example 7. Raise −5 to the third power.

The number −5 to the third power is the product of three factors, each of which is equal to −5. Exponent 3 is an odd number, so we can say in advance that the answer will be negative:

(−5) 3 = (−5) × (−5) × (−5) = −125

Example 8. Raise −4 to the fourth power.

The number −4 to the fourth power is the product of four factors, each of which is equal to −4. Moreover, exponent 4 is even, so we can say in advance that the answer will be positive:

(−4) 4 = (−4) × (−4) × (−4) × (−4) = 256

Finding Expression Values

When finding the values of expressions that do not contain parentheses, exponentiation will be performed first, followed by multiplication and division in the order they appear, and then addition and subtraction in the order they appear.

Example 1. Find the value of the expression 2 + 5 2

First, exponentiation is performed. In this case, the number 5 is raised to the second power - we get 25. Then this result is added to the number 2

2 + 5 2 = 2 + 25 = 27

Example 10. Find the value of the expression −6 2 × (−12)

First, exponentiation is performed. Note that the number −6 is not in parentheses, so the number 6 will be raised to the second power, then a minus will be placed in front of the result:

−6 2 × (−12) = −36 × (−12)

We complete the example by multiplying −36 by (−12)

−6 2 × (−12) = −36 × (−12) = 432

Example 11. Find the value of the expression −3 × 2 2

First, exponentiation is performed. Then the resulting result is multiplied with the number −3

−3 × 2 2 = −3 × 4 = −12

If the expression contains parentheses, then you must first perform the operations in these parentheses, then exponentiation, then multiplication and division, and then addition and subtraction.

Example 12. Find the value of the expression (3 2 + 1 × 3) − 15 + 5

First we perform the actions in brackets. Inside the brackets, we apply the previously learned rules, namely, first we raise the number 3 to the second power, then we multiply 1 × 3, then we add the results of raising the number 3 to the second power and multiplying 1 × 3. Next, subtraction and addition are performed in the order they appear. Let's arrange the following order of executing the action on the original expression:

(3 2 + 1 × 3) − 15 + 5 = 12 − 15 + 5 = 2

Example 13. Find the value of the expression 2 × 5 3 + 5 × 2 3

First, let's raise the numbers to powers, then multiply and add the results:

2 × 5 3 + 5 × 2 3 = 2 × 125 + 5 × 8 = 250 + 40 = 290

Identical power transformations

Various identity transformations can be performed on powers, thereby simplifying them.

Let's say we needed to calculate the expression (2 3) 2. In this example, two to the third power is raised to the second power. In other words, a degree is raised to another degree.

(2 3) 2 is the product of two powers, each of which is equal to 2 3

Moreover, each of these powers is the product of three factors, each of which is equal to 2

We got the product 2 × 2 × 2 × 2 × 2 × 2, which is equal to 64. This means the value of the expression (2 3) 2 or equal to 64

This example can be greatly simplified. To do this, the exponents of the expression (2 3) 2 can be multiplied and this product written over the base 2

We received 2 6. Two to the sixth power is the product of six factors, each of which is equal to 2. This product is equal to 64

This property works because 2 3 is the product of 2 × 2 × 2, which in turn is repeated twice. Then it turns out that base 2 is repeated six times. From here we can write that 2 × 2 × 2 × 2 × 2 × 2 is 2 6

In general, for any reason a with indicators m And n, the following equality holds:

(a n)m = a n × m

This identical transformation is called raising a power to a power. It can be read like this: “When raising a power to a power, the base is left unchanged, and the exponents are multiplied” .

After multiplying the indicators, you get another degree, the value of which can be found.

Example 2. Find the value of the expression (3 2) 2

In this example, the base is 3, and the numbers 2 and 2 are exponents. Let's use the rule of raising a power to a power. We will leave the base unchanged, and multiply the indicators:

We got 3 4. And the number 3 to the fourth power is 81

Let's consider the remaining transformations.

Multiplying powers

To multiply powers, you need to separately calculate each power and multiply the results.

For example, let's multiply 2 2 by 3 3.

2 2 is the number 4, and 3 3 is the number 27. Multiply the numbers 4 and 27, we get 108

2 2 × 3 3 = 4 × 27 = 108

In this example, the degree bases were different. If the bases are the same, then you can write down one base, and write down the sum of the indicators of the original degrees as an indicator.

For example, multiply 2 2 by 2 3

In this example, the bases for the degrees are the same. In this case, you can write down one base 2 and write down the sum of the exponents of powers 2 2 and 2 3 as an exponent. In other words, leave the basis unchanged, and add up the indicators of the original degrees. It will look like this:

We received 2 5. The number 2 to the fifth power is 32

This property works because 2 2 is the product of 2 × 2, and 2 3 is the product of 2 × 2 × 2. Then we get a product of five identical factors, each of which is equal to 2. This product can be represented as 2 5

In general, for anyone a and indicators m And n the following equality holds:

This identical transformation is called basic property of degree. It can be read like this: " PWhen multiplying powers with the same bases, the base is left unchanged, and the exponents are added.” .

Note that this transformation can be applied to any number of degrees. The main thing is that the base is the same.

For example, let's find the value of the expression 2 1 × 2 2 × 2 3. Base 2

In some problems, it may be sufficient to perform the appropriate transformation without calculating the final degree. This is of course very convenient, since calculating large powers is not so easy.

Example 1. Express as a power the expression 5 8 × 25

In this problem, you need to make sure that instead of the expression 5 8 × 25, you get one power.

The number 25 can be represented as 5 2. Then we get the following expression:

In this expression, you can apply the basic property of the degree - leave the base 5 unchanged, and add the exponents 8 and 2:

Let's write down the solution briefly:

Example 2. Express as a power the expression 2 9 × 32

The number 32 can be represented as 2 5. Then we get the expression 2 9 × 2 5. Next, you can apply the base property of the degree - leave base 2 unchanged, and add exponents 9 and 5. The result will be the following solution:

Example 3. Calculate the 3 × 3 product using the basic property of powers.

Everyone knows well that three times three equals nine, but the problem requires using the basic property of degrees in the solution. How to do it?

We recall that if a number is given without an indicator, then the indicator must be considered equal to one. Therefore, factors 3 and 3 can be written as 3 1 and 3 1

3 1 × 3 1

Now let's use the basic property of degree. We leave base 3 unchanged, and add up indicators 1 and 1:

3 1 × 3 1 = 3 2 = 9

Example 4. Calculate the product 2 × 2 × 3 2 × 3 3 using the basic property of powers.

We replace the product 2 × 2 with 2 1 × 2 1, then with 2 1 + 1, and then with 2 2. Replace the product 3 2 × 3 3 with 3 2 + 3 and then with 3 5

Example 5. Perform multiplication x × x

These are two identical letter factors with exponents 1. For clarity, let’s write down these exponents. Next is the base x Let's leave it unchanged and add up the indicators:

While at the board, you should not write down the multiplication of powers with the same bases in as much detail as is done here. Such calculations must be done in your head. A detailed note will most likely irritate the teacher and he will reduce the grade for it. Here, a detailed recording is given to make the material as easy to understand as possible.

It is advisable to write the solution to this example as follows:

Example 6. Perform multiplication x 2 × x

The exponent of the second factor is equal to one. For clarity, let's write it down. Next, we will leave the base unchanged and add up the indicators:

Example 7. Perform multiplication y 3 y 2 y

The exponent of the third factor is equal to one. For clarity, let's write it down. Next, we will leave the base unchanged and add up the indicators:

Example 8. Perform multiplication aa 3 a 2 a 5

The exponent of the first factor is equal to one. For clarity, let's write it down. Next, we will leave the base unchanged and add up the indicators:

Example 9. Represent the power 3 8 as a product of powers with the same bases.

In this problem, you need to create a product of powers whose bases will be equal to 3, and the sum of whose exponents will be equal to 8. Any indicators can be used. Let us represent the power 3 8 as the product of the powers 3 5 and 3 3

In this example, we again relied on the basic property of degree. After all, the expression 3 5 × 3 3 can be written as 3 5 + 3, from which 3 8.

Of course, it was possible to represent the power 3 8 as a product of other powers. For example, in the form 3 7 × 3 1, since this product is also equal to 3 8

Representing a degree as a product of powers with the same bases is mostly a creative work. Therefore, there is no need to be afraid to experiment.

Example 10. Submit Degree x 12 in the form of various products of powers with bases x .

Let's use the basic property of degrees. Let's imagine x 12 in the form of products with bases x, and the sum of the indicators is 12

Constructs with sums of indicators were recorded for clarity. Most often you can skip them. Then you get a compact solution:

Raising to the power of a product

To raise a product to a power, you need to raise each factor of this product to the specified power and multiply the results.

For example, let's raise the product 2 × 3 to the second power. Let's take this product in brackets and indicate 2 as an indicator

Now let’s raise each factor of the 2 × 3 product to the second power and multiply the results:

The principle of operation of this rule is based on the definition of degree, which was given at the very beginning.

Raising the product 2 × 3 to the second power means repeating the product twice. And if you repeat it twice, you can get the following:

2 × 3 × 2 × 3

Rearranging the places of the factors does not change the product. This allows you to group like factors:

2 × 2 × 3 × 3

Repeating factors can be replaced with short entries - bases with indicators. The product 2 × 2 can be replaced by 2 2, and the product 3 × 3 can be replaced by 3 2. Then the expression 2 × 2 × 3 × 3 becomes the expression 2 2 × 3 2.

Let ab original work. To raise a given product to a power n, you need to multiply the factors separately a And b to the specified degree n

This property is true for any number of factors. The following expressions are also valid:

Example 2. Find the value of the expression (2 × 3 × 4) 2

In this example, you need to raise the product 2 × 3 × 4 to the second power. To do this, you need to raise each factor of this product to the second power and multiply the results:

Example 3. Raise the product to the third power a×b×c

Let us enclose this product in brackets and indicate the number 3 as an indicator

Example 4. Raise the product 3 to the third power xyz

Let us enclose this product in brackets and indicate 3 as an indicator

(3xyz) 3

Let us raise each factor of this product to the third power:

(3xyz) 3 = 3 3 x 3 y 3 z 3

The number 3 to the third power is equal to the number 27. We will leave the rest unchanged:

(3xyz) 3 = 3 3 x 3 y 3 z 3 = 27x 3 y 3 z 3

In some examples, multiplication of powers with the same exponents can be replaced by the product of bases with the same exponent.

For example, let's calculate the value of the expression 5 2 × 3 2. Let's raise each number to the second power and multiply the results:

5 2 × 3 2 = 25 × 9 = 225

But you don't have to calculate each degree separately. Instead, this product of powers can be replaced by a product with one exponent (5 × 3) 2 . Next, calculate the value in parentheses and raise the result to the second power:

5 2 × 3 2 = (5 × 3) 2 = (15) 2 = 225

In this case, the rule of exponentiation of a product was again used. After all, if (a×b)n = a n × b n , That a n × b n = (a × b)n. That is, the left and right sides of the equality have swapped places.

Raising a degree to a power

We considered this transformation as an example when we tried to understand the essence of identical transformations of degrees.

When raising a power to a power, the base is left unchanged, and the exponents are multiplied:

(a n)m = a n × m

For example, the expression (2 3) 2 is a power raised to a power - two to the third power is raised to the second power. To find the value of this expression, the base can be left unchanged and the exponents can be multiplied:

(2 3) 2 = 2 3 × 2 = 2 6

(2 3) 2 = 2 3 × 2 = 2 6 = 64

This rule is based on the previous rules: exponentiation of the product and the basic property of the degree.

Let's return to the expression (2 3) 2. The expression in brackets 2 3 is a product of three identical factors, each of which is equal to 2. Then in the expression (2 3) the 2 power inside the brackets can be replaced by the product 2 × 2 × 2.

(2 × 2 × 2) 2

And this is the exponentiation of the product that we studied earlier. Let us recall that to raise a product to a power, you need to raise each factor of a given product to the indicated power and multiply the results obtained:

(2 × 2 × 2) 2 = 2 2 × 2 2 × 2 2

Now we are dealing with the basic property of degree. We leave the base unchanged and add up the indicators:

(2 × 2 × 2) 2 = 2 2 × 2 2 × 2 2 = 2 2 + 2 + 2 = 2 6

As before, we received 2 6. The value of this degree is 64

(2 × 2 × 2) 2 = 2 2 × 2 2 × 2 2 = 2 2 + 2 + 2 = 2 6 = 64

A product whose factors are also powers can also be raised to a power.

For example, let's find the value of the expression (2 2 × 3 2) 3. Here, the indicators of each multiplier must be multiplied by the total indicator 3. Next, find the value of each degree and calculate the product:

(2 2 × 3 2) 3 = 2 2 × 3 × 3 2 × 3 = 2 6 × 3 6 = 64 × 729 = 46656

Approximately the same thing happens when raising a product to a power. We said that when raising a product to a power, each factor of this product is raised to the indicated power.

For example, to raise the product 2 × 4 to the third power, you would write the following expression:

But earlier it was said that if a number is given without an indicator, then the indicator must be considered equal to one. It turns out that the factors of the product 2 × 4 initially have exponents equal to 1. This means that the expression 2 1 × 4 1 was raised to the third power. And this is raising a degree to a degree.

Let's rewrite the solution using the rule for raising a power to a power. We should get the same result:

Example 2. Find the value of the expression (3 3) 2

We leave the base unchanged, and multiply the indicators:

We got 3 6. The number 3 to the sixth power is the number 729

Example 3xy)³

Example 4. Perform exponentiation in the expression ( abc)⁵

Let us raise each factor of the product to the fifth power:

Example 5ax) 3

Let us raise each factor of the product to the third power:

Since negative number −2 was raised to the third power, it was placed in parentheses.

Example 6. Perform exponentiation in expression (10 xy) 2

Example 7. Perform exponentiation in the expression (−5 x) 3

Example 8. Perform exponentiation in the expression (−3 y) 4

Example 9. Perform exponentiation in the expression (−2 abx)⁴

Example 10. Simplify the expression x 5×( x 2) 3

Degree x Let us leave 5 unchanged for now, and in the expression ( x 2) 3 we perform the raising of a power to a power:

x 5 × (x 2) 3 = x 5 × x 2×3 = x 5 × x 6

Now let's do the multiplication x 5 × x 6. To do this, we will use the basic property of a degree - the base x Let's leave it unchanged and add up the indicators:

x 5 × (x 2) 3 = x 5 × x 2×3 = x 5 × x 6 = x 5 + 6 = x 11

Example 9. Find the value of the expression 4 3 × 2 2 using the basic property of power.

The basic property of a degree can be used if the bases of the original degrees are the same. In this example, the bases are different, so first you need to modify the original expression a little, namely, make sure that the bases of the powers become the same.

Let's look closely at the degree 4 3. The base of this degree is the number 4, which can be represented as 2 2. Then the original expression will take the form (2 2) 3 × 2 2. By raising the power to the power in the expression (2 2) 3, we get 2 6. Then the original expression will take the form 2 6 × 2 2, which can be calculated using the basic property of power.

Let's write down the solution to this example:

Division of degrees

To perform division of powers, you need to find the value of each power, then divide ordinary numbers.

For example, let's divide 4 3 by 2 2.

Let's calculate 4 3, we get 64. Calculate 2 2, get 4. Now divide 64 by 4, get 16

If, when dividing the powers, the bases turn out to be the same, then the base can be left unchanged, and the exponent of the divisor can be subtracted from the exponent of the dividend.

For example, let's find the value of the expression 2 3: 2 2

We leave base 2 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

This means that the value of the expression 2 3: 2 2 is equal to 2.

This property is based on the multiplication of powers with the same bases, or, as we used to say, the basic property of a power.

Let's return to the previous example 2 3: 2 2. Here the dividend is 2 3 and the divisor is 2 2.

Dividing one number by another means finding a number that, when multiplied by the divisor, will result in the dividend.

In our case, dividing 2 3 by 2 2 means finding a power that, when multiplied by the divisor 2 2, results in 2 3. What power can be multiplied by 2 2 to get 2 3? Obviously, only the degree 2 is 1. From the basic property of degree we have:

You can verify that the value of the expression 2 3: 2 2 is equal to 2 1 by directly calculating the expression 2 3: 2 2 itself. To do this, we first find the value of the power 2 3, we get 8. Then we find the value of the power 2 2, we get 4. Divide 8 by 4, we get 2 or 2 1, since 2 = 2 1.

2 3: 2 2 = 8: 4 = 2

Thus, when dividing powers with the same bases, the following equality holds:

It may also happen that not only the reasons, but also the indicators may be the same. In this case, the answer will be one.

For example, let's find the value of the expression 2 2: 2 2. Let's calculate the value of each degree and divide the resulting numbers:

When solving example 2 2: 2 2, you can also apply the rule of dividing powers with the same bases. The result is a number to the zero power, since the difference between the exponents of the powers 2 2 and 2 2 is equal to zero:

We found out above why the number 2 to the zero power is equal to one. If you calculate 2 2: 2 2 using the usual method, without using the power division rule, you get one.

Example 2. Find the value of the expression 4 12: 4 10

Let us leave 4 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

4 12: 4 10 = 4 12 − 10 = 4 2 = 16

Example 3. Present the quotient x 3: x in the form of a power with a base x

Let's use the power division rule. Base x Let's leave it unchanged, and subtract the exponent of the divisor from the exponent of the dividend. The divisor exponent is equal to one. For clarity, let's write it down:

Example 4. Present the quotient x 3: x 2 as a power with a base x

Let's use the power division rule. Base x

Division of powers can be written as a fraction. So, the previous example can be written as follows:

The numerator and denominator of a fraction can be written in expanded form, namely in the form of products of identical factors. Degree x 3 can be written as x × x × x, and the degree x 2 how x × x. Then the design x 3 − 2 can be skipped and the fraction can be reduced. It will be possible to reduce two factors in the numerator and denominator x. As a result, one multiplier will remain x

Or even shorter:

It is also useful to be able to quickly reduce fractions consisting of powers. For example, a fraction can be reduced by x 2. To reduce a fraction by x 2 you need to divide the numerator and denominator of the fraction by x 2

The division of degrees need not be described in detail. The above abbreviation can be done shorter:

Or even shorter:

Example 5. Perform division x 12 : x 3

Let's use the power division rule. Base x leave it unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

Let's write the solution using fraction reduction. Division of degrees x 12 : x Let's write 3 in the form . Next, we reduce this fraction by x 3 .

Example 6. Find the value of an expression

In the numerator we perform multiplication of powers with the same bases:

Now we apply the rule for dividing powers with the same bases. We leave base 7 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

We complete the example by calculating the power 7 2

Example 7. Find the value of an expression

Let's raise the power to the power in the numerator. You need to do this with the expression (2 3) 4

Now let's multiply powers with the same bases in the numerator.

Addition and subtraction of powers

It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds equal powers of identical variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is equal to 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3.

It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 — 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Multiplying powers

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.

Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the multiplication result, which is equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n;

And a m is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents of the powers.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x – 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y -n .y -m = y -n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.

So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.

Division of degrees

Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.

Thus, a 3 b 2 divided by b 2 is equal to a 3.

Writing a 5 divided by a 3 looks like $\frac $. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing degrees with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1. That is, $\frac = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.

Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3

The rule is also true for numbers with negative values of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$

It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Decrease the exponents by $\frac $ Answer: $\frac $.

2. Decrease exponents by $\frac$. Answer: $\frac$ or 2x.

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

Properties of degree

We remind you that in this lesson we will understand properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.

A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.

Property No. 1
Product of powers

When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.

a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.

This property of powers also applies to the product of three or more powers.

Simplify the expression.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
Present it as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
Present it as a degree.
(0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Please note that in the specified property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243

Property No. 2
Partial degrees

When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
Calculate.

11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4

Answer: t = 3 4 = 81

Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

Example. Find the value of an expression using the properties of exponents.

2 11 − 5 = 2 6 = 64

Please note that in Property 2 we were only talking about dividing powers with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4

Property No. 3
Raising a degree to a power

When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.

We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

How to multiply powers

How to multiply powers? Which powers can be multiplied and which cannot? How to multiply a number by a power?

In algebra, you can find a product of powers in two cases:

1) if the degrees have the same bases;

2) if the degrees have the same indicators.

When multiplying powers with the same bases, the base must be left the same, and the exponents must be added:

When multiplying degrees with the same indicators, the overall indicator can be taken out of brackets:

Let's look at how to multiply powers using specific examples.

The unit is not written in the exponent, but when multiplying powers, they take into account:

When multiplying, there can be any number of powers. It should be remembered that you don’t have to write the multiplication sign before the letter:

In expressions, exponentiation is done first.

If you need to multiply a number by a power, you should first perform the exponentiation, and only then the multiplication:

Multiplying powers with the same bases

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In this lesson we will study multiplication of powers with like bases. First, let us recall the definition of degree and formulate a theorem on the validity of the equality . Then we will give examples of its application on specific numbers and prove it. We will also apply the theorem to solve various problems.

Topic: Power with a natural exponent and its properties

Lesson: Multiplying powers with the same bases (formula)

1. Basic definitions

Basic definitions:

n- exponent,

— n th power of a number.

2. Statement of Theorem 1

Theorem 1. For any number A and any natural n And k the equality is true:

In other words: if A– any number; n And k natural numbers, then:

Hence rule 1:

3. Explanatory tasks

Conclusion: special cases confirmed the correctness of Theorem No. 1. Let us prove it in the general case, that is, for any A and any natural n And k.

4. Proof of Theorem 1

Given a number A– any; numbers n And k – natural. Prove:

The proof is based on the definition of degree.

5. Solving examples using Theorem 1

Example 1: Think of it as a degree.

To solve the following examples, we will use Theorem 1.

and)

6. Generalization of Theorem 1

A generalization used here:

7. Solving examples using a generalization of Theorem 1

8. Solving various problems using Theorem 1

Example 2: Calculate (you can use the table of basic powers).

A) (according to the table)

Example 3: Write it as a power with base 2.

Example 4: Determine the sign of the number:

, A - negative, since the exponent at -13 is odd.

Example 5: Replace (·) with a power of a number with a base r:

We have, that is.

9. Summing up

1. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 7. 6th edition. M.: Enlightenment. 2010

1. School assistant (Source).

1. Present as a power:

a B C D E)

3. Write as a power with base 2:

4. Determine the sign of the number:

5. Replace (·) with a power of a number with a base r:

a) r 4 · (·) = r 15; b) (·) · r 5 = r 6

Multiplication and division of powers with the same exponents

In this lesson we will study multiplication of powers with equal exponents. First, let's recall the basic definitions and theorems about multiplying and dividing powers with the same bases and raising powers to powers. Then we formulate and prove theorems on multiplication and division of powers with the same exponents. And then with their help we will solve a number of typical problems.

Reminder of basic definitions and theorems

Here a- the basis of the degree,

— n th power of a number.

Theorem 1. For any number A and any natural n And k the equality is true:

When multiplying powers with the same bases, the exponents are added, the base remains unchanged.

Theorem 2. For any number A and any natural n And k, such that n > k the equality is true:

When dividing degrees with the same bases, the exponents are subtracted, but the base remains unchanged.

Theorem 3. For any number A and any natural n And k the equality is true:

All the theorems listed were about powers with the same reasons, in this lesson we will look at degrees with the same indicators.

Examples for multiplying powers with the same exponents

Consider the following examples:

Let's write down the expressions for determining the degree.

Conclusion: From the examples it can be seen that , but this still needs to be proven. Let us formulate the theorem and prove it in the general case, that is, for any A And b and any natural n.

Formulation and proof of Theorem 4

For any numbers A And b and any natural n the equality is true:

Proof Theorem 4 .

By definition of degree:

So we have proven that .

To multiply powers with the same exponents, it is enough to multiply the bases and leave the exponent unchanged.

Formulation and proof of Theorem 5

Let us formulate a theorem for dividing powers with the same exponents.

For any number A And b() and any natural n the equality is true:

Proof Theorem 5 .

Let's write down the definition of degree:

Statement of theorems in words

So, we have proven that .

To divide powers with the same exponents into each other, it is enough to divide one base by another, and leave the exponent unchanged.

Solving typical problems using Theorem 4

Example 1: Present as a product of powers.

To solve the following examples, we will use Theorem 4.

To solve the following example, recall the formulas:

Generalization of Theorem 4

Generalization of Theorem 4:

Solving Examples Using Generalized Theorem 4

Continuing to solve typical problems

Example 2: Write it as a power of the product.

Example 3: Write it as a power with exponent 2.

Calculation examples

Example 4: Calculate in the most rational way.

2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: VENTANA-GRAF

3. Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E. and others. Algebra 7.M.: Enlightenment. 2006

2. School assistant (Source).

1. Present as a product of powers:

A) ; b) ; V) ; G) ;

2. Write as a power of the product:

3. Write as a power with exponent 2:

4. Calculate in the most rational way.

Mathematics lesson on the topic “Multiplication and division of powers”

Sections: Mathematics

Pedagogical goal:

the student will learn distinguish between the properties of multiplication and division of powers with natural exponents; apply these properties in the case of the same bases;

the student will have the opportunity be able to perform transformations of degrees with different bases and be able to perform transformations in combined tasks.

Tasks:

organize students’ work by repeating previously studied material;

ensure the level of reproduction by performing various types of exercises;

organize a check on students’ self-assessment through testing.

Activity units of teaching: determination of degree with a natural indicator; degree components; definition of private; combinational law of multiplication.

I. Organizing a demonstration of students’ mastery of existing knowledge. (step 1)

a) Updating knowledge:

2) Formulate a definition of degree with a natural exponent.

a n =a a a a … a (n times)

b k =b b b b a… b (k times) Justify the answer.

II. Organization of self-assessment of the student’s degree of proficiency in current experience. (step 2)

Self-test: (individual work in two versions.)

A1) Present the product 7 7 7 7 x x x as a power:

A2) Represent the power (-3) 3 x 2 as a product

A3) Calculate: -2 3 2 + 4 5 3

I select the number of tasks in the test in accordance with the preparation of the class level.

I give you the key to the test for self-test. Criteria: pass - no pass.

III. Educational and practical task (step 3) + step 4. (the students themselves will formulate the properties)

calculate: 2 2 2 3 = ? 3 3 3 2 3 =?

Simplify: a 2 a 20 = ? b 30 b 10 b 15 = ?

While solving problems 1) and 2), students propose a solution, and I, as a teacher, organize the class to find a way to simplify powers when multiplying with the same bases.

Teacher: come up with a way to simplify powers when multiplying with the same bases.

An entry appears on the cluster:

The topic of the lesson is formulated. Multiplication of powers.

Teacher: come up with a rule for dividing powers with the same bases.

Reasoning: what action is used to check division? a 5: a 3 = ? that a 2 a 3 = a 5

I return to the diagram - a cluster and add to the entry - .. when dividing, we subtract and add the topic of the lesson. ...and division of degrees.

IV. Communicating to students the limits of knowledge (as a minimum and as a maximum).

Teacher: the minimum task for today’s lesson is to learn to apply the properties of multiplication and division of powers with the same bases, and the maximum task is to apply multiplication and division together.

We write on the board : a m a n = a m+n ; a m: a n = a m-n

V. Organization of studying new material. (step 5)

a) According to the textbook: No. 403 (a, c, e) tasks with different wordings

No. 404 (a, d, f) independent work, then I organize a mutual check, give the keys.

b) For what value of m is the equality valid? a 16 a m = a 32; x h x 14 = x 28; x 8 (*) = x 14

Assignment: come up with similar examples for division.

c) No. 417 (a), No. 418 (a) Traps for students: x 3 x n = x 3n; 3 4 3 2 = 9 6 ; a 16: a 8 = a 2.

VI. Summarizing what has been learned, conducting diagnostic work (which encourages students, and not the teacher, to study this topic) (step 6)

Diagnostic work.

Test(place the keys on the back of the dough).

Task options: represent the quotient x 15 as a power: x 3; represent as a power the product (-4) 2 (-4) 5 (-4) 7 ; for which m is the equality a 16 a m = a 32 valid? find the value of the expression h 0: h 2 at h = 0.2; calculate the value of the expression (5 2 5 0) : 5 2 .

Lesson summary. Reflection. I divide the class into two groups.

Find arguments in group I: in favor of knowing the properties of the degree, and group II - arguments that will say that you can do without properties. We listen to all the answers and draw conclusions. In subsequent lessons, you can offer statistical data and call the rubric “It’s beyond belief!”

The average person eats 32 10 2 kg of cucumbers during their lifetime.

The wasp is capable of making a non-stop flight of 3.2 10 2 km.

When glass cracks, the crack propagates at a speed of about 5 10 3 km/h.

A frog eats more than 3 tons of mosquitoes in its life. Using the degree, write in kg.

The most prolific is considered to be the ocean fish - the moon (Mola mola), which lays up to 300,000,000 eggs with a diameter of about 1.3 mm in one spawning. Write this number using a power.

VII. Homework.

Historical reference. What numbers are called Fermat numbers.

P.19. No. 403, No. 408, No. 417

Used Books:

Textbook "Algebra-7", authors Yu.N. Makarychev, N.G. Mindyuk et al.

Didactic material for 7th grade, L.V. Kuznetsova, L.I. Zvavich, S.B. Suvorov.

Encyclopedia of mathematics.

Magazine "Kvant".

Properties of degrees, formulations, proofs, examples.

After the power of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.

Page navigation.

Properties of degrees with natural exponents

By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, we can obtain and justify the following properties of degree with natural exponent:

the main property of the degree a m ·a n =a m+n, its generalization a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k;

property of quotient powers with identical bases a m:a n =a m−n ;

property of the degree of a product (a·b) n =a n ·b n , its extension (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n ;

property of the quotient to the natural degree (a:b) n =a n:b n ;

raising a degree to a power (a m) n =a m·n, its generalization (((a n 1) n 2) …) n k =a n 1 ·n 2 ·…·n k;

comparison of degree with zero:

if a>0, then a n>0 for any natural number n;
if a=0, then a n =0;
if a 2·m >0 , if a 2·m−1 n ;
if m and n are natural numbers such that m>n, then for 0m n, and for a>0 the inequality a m >a n is true.

Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying expressions often used in the form a m+n =a m ·a n .

Now let's look at each of them in detail.

Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.

Let us prove the main property of the degree. By the definition of a power with a natural exponent, the product of powers with identical bases of the form a m ·a n can be written as the product . Due to the properties of multiplication, the resulting expression can be written as , and this product is a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.

Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values of the expressions 2 2 · 2 3 and 2 5 . Carrying out exponentiation, we have 2 2 2 3 =(2 2) (2 2 2) = 4 8 = 32 and 2 5 =2 2 2 2 2 = 32 , since we get equal values, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.

The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1 , n 2 , …, n k the equality a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k is true.

For example, (2,1) 3 ·(2,1) 3 ·(2,1) 4 ·(2,1) 7 = (2,1) 3+3+4+7 =(2,1) 17.

We can move on to the next property of powers with a natural exponent – property of quotient powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n, the equality a m:a n =a m−n is true.

Before presenting the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that we cannot divide by zero. The condition m>n is introduced so that we do not go beyond the natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n) or a negative number (which happens for m m−n ·a n =a (m−n) +n =a m. From the resulting equality a m−n ·a n =a m and from the connection between multiplication and division it follows that a m−n is a quotient of powers a m and an n. This proves the property of quotients of powers with the same bases.

Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the equality π 5:π 2 =π 5−3 =π 3 corresponds to the considered property of the degree.

Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n .

Indeed, by the definition of a degree with a natural exponent we have . Based on the properties of multiplication, the last product can be rewritten as , which is equal to a n · b n .

Here's an example: .

This property extends to the power of the product of three or more factors. That is, the property of natural degree n of a product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n .

For clarity, we will show this property with an example. For the product of three factors to the power of 7 we have .

The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of powers a n and b n, that is, (a:b) n =a n:b n.

The proof can be carried out using the previous property. So (a:b) n ·b n =((a:b)·b) n =a n , and from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of division a n on bn.

Let's write this property using specific numbers as an example: .

Now let's voice it property of raising a power to a power: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of the number a with exponent m·n, that is, (a m) n =a m·n.

For example, (5 2) 3 =5 2·3 =5 6.

The proof of the power-to-degree property is the following chain of equalities: .

The property considered can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality . For greater clarity, let's give an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10.

It remains to dwell on the properties of comparing degrees with a natural exponent.

Let's start by proving the property of comparing zero and power with a natural exponent.

First, let's prove that a n >0 for any a>0.

The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a, the degree a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and .

It is quite obvious that for any natural number n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0.

Let's move on to negative bases of degree.

Let's start with the case when the exponent is an even number, let's denote it as 2·m, where m is a natural number. Then . According to the rule for multiplying negative numbers, each of the products of the form a·a is equal to the product of the absolute values of the numbers a and a, which means that it is a positive number. Therefore, the product will also be positive and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .

Finally, when the base a is a negative number and the exponent is an odd number 2 m−1, then . All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3 17 n n is the product of the left and right sides of n true inequalities a properties of inequalities, a provable inequality of the form a n n is also true. For example, due to this property, the inequalities 3 7 7 and .

It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of two powers with natural exponents and identical positive bases less than one, the one whose exponent is smaller is greater; and of two powers with natural exponents and identical bases greater than one, the one whose exponent is greater is greater. Let us proceed to the proof of this property.

Let us prove that for m>n and 0m n . To do this, we write down the difference a m − a n and compare it with zero. The recorded difference, after taking a n out of brackets, will take the form a n ·(a m−n−1) . The resulting product is negative as the product of a positive number a n and a negative number a m−n −1 (a n is positive as the natural power of a positive number, and the difference a m−n −1 is negative, since m−n>0 due to the initial condition m>n, whence it follows that when 0m−n is less than unity). Therefore, a m −a n m n , which is what needed to be proven. As an example, we give the correct inequality.

It remains to prove the second part of the property. Let us prove that for m>n and a>1 a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1 the degree a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.

Properties of powers with integer exponents

Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.

We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.

So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true: properties of powers with integer exponents:

a m ·a n =a m+n ;
a m:a n =a m−n ;
(a·b) n =a n ·b n ;
(a:b) n =a n:b n ;
(a m) n =a m·n ;
if n is a positive integer, a and b are positive numbers, and a n n and a −n >b −n ;
if m and n are integers, and m>n, then for 0m n, and for a>1 the inequality a m >a n holds.

When a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.

Proving each of these properties is not difficult; to do this, it is enough to use the definitions of degrees with natural and integer exponents, as well as the properties of operations with real numbers. As an example, let us prove that the power-to-power property holds for both positive integers and non-positive integers. To do this, you need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (a p) q =a p·q, (a −p) q =a (−p)·q, (a p ) −q =a p·(−q) and (a −p) −q =a (−p)·(−q) . Let's do it.

For positive p and q, the equality (a p) q =a p·q was proven in the previous paragraph. If p=0, then we have (a 0) q =1 q =1 and a 0·q =a 0 =1, whence (a 0) q =a 0·q. Similarly, if q=0, then (a p) 0 =1 and a p·0 =a 0 =1, whence (a p) 0 =a p·0. If both p=0 and q=0, then (a 0) 0 =1 0 =1 and a 0·0 =a 0 =1, whence (a 0) 0 =a 0·0.

Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then . By the property of quotients to powers we have . Since 1 p =1·1·…·1=1 and , then . The last expression, by definition, is a power of the form a −(p·q), which, due to the rules of multiplication, can be written as a (−p)·q.

Likewise .

AND .

Using the same principle, you can prove all other properties of a degree with an integer exponent, written in the form of equalities.

In the penultimate of the recorded properties, it is worth dwelling on the proof of the inequality a −n >b −n, which is valid for any negative integer −n and any positive a and bfor which the condition a is satisfied . Let us write down and transform the difference between the left and right sides of this inequality: . Since by condition a n n , therefore, b n −a n >0 . The product a n · b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as the quotient of the positive numbers b n −a n and a n ·b n . Therefore, whence a −n >b −n , which is what needed to be proved.

The last property of powers with integer exponents is proven in the same way as a similar property of powers with natural exponents.

Properties of powers with rational exponents

We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers with integer exponents. Namely:

property of the product of powers with the same bases for a>0, and if and, then for a≥0;
property of quotient powers with the same bases for a>0 ;
property of a product to a fractional power for a>0 and b>0, and if and, then for a≥0 and (or) b≥0;
property of a quotient to a fractional power for a>0 and b>0, and if , then for a≥0 and b>0;
property of degree to degree for a>0, and if and, then for a≥0;
property of comparing powers with equal rational exponents: for any positive numbers a and b, a 0 the inequality a p p is true, and for p p >b p ;
the property of comparing powers with rational exponents and equal bases: for rational numbers p and q, p>q for 0p q, and for a>0 – inequality a p >a q.

The proof of the properties of powers with fractional exponents is based on the definition of a power with a fractional exponent, on the properties of the arithmetic root of the nth degree and on the properties of a power with an integer exponent. Let us provide evidence.

By definition of a power with a fractional exponent and , then . The properties of the arithmetic root allow us to write the following equalities. Further, using the property of a degree with an integer exponent, we obtain , from which, by the definition of a degree with a fractional exponent, we have , and the indicator of the degree obtained can be transformed as follows: . This completes the proof.

The second property of powers with fractional exponents is proved in an absolutely similar way:

The remaining equalities are proved using similar principles:

Let's move on to proving the next property. Let us prove that for any positive a and b, a 0 the inequality a p p is true, and for p p >b p . Let's write the rational number p as m/n, where m is an integer and n is a natural number. The conditions p 0 in this case will be equivalent to the conditions m 0, respectively. For m>0 and am m . From this inequality, by the property of roots, we have, and since a and b are positive numbers, then, based on the definition of a degree with a fractional exponent, the resulting inequality can be rewritten as, that is, a p p .

Similarly, for m m >b m , whence, that is, a p >b p .

It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q, p>q for 0p q, and for a>0 – the inequality a p >a q. We can always reduce rational numbers p and q to a common denominator, even if we get ordinary fractions and , where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from the rule for comparing ordinary fractions with the same denominators. Then, by the property of comparing degrees with the same bases and natural exponents, for 0m 1 m 2, and for a>1, the inequality a m 1 >a m 2. These inequalities in the properties of the roots can be rewritten accordingly as And . And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From here we draw the final conclusion: for p>q and 0p q , and for a>0 – the inequality a p >a q .

Properties of powers with irrational exponents

From the way a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0, b>0 and irrational numbers p and q the following are true properties of powers with irrational exponents:

a p ·a q =a p+q ;
a p:a q =a p−q ;
(a·b) p =a p ·b p ;
(a:b) p =a p:b p ;
(a p) q =a p·q ;
for any positive numbers a and b, a 0 the inequality a p p is true, and for p p >b p ;
for irrational numbers p and q, p>q for 0p q, and for a>0 – the inequality a p >a q.

From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.

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In the previous article we explained what monomials are. In this material we will look at how to solve examples and problems in which they are used. Here we will consider such actions as subtraction, addition, multiplication, division of monomials and raising them to a power with a natural exponent. We will show how such operations are defined, outline the basic rules for their implementation and what should be the result. All theoretical concepts, as usual, will be illustrated with examples of problems with descriptions of solutions.

It is most convenient to work with the standard notation of monomials, so we present all expressions that will be used in the article in standard form. If they were originally specified differently, it is recommended to first bring them to a generally accepted form.

Rules for adding and subtracting monomials

The simplest operations that can be performed with monomials are subtraction and addition. In general, the result of these actions will be a polynomial (a monomial is possible in some special cases).

When we add or subtract monomials, we first write down the corresponding sum and difference in the generally accepted form, and then simplify the resulting expression. If there are similar terms, they need to be cited, and the parentheses should be opened. Let's explain with an example.

Example 1

Condition: perform the addition of the monomials − 3 x and 2, 72 x 3 y 5 z.

Solution

Let's write down the sum of the original expressions. Let's add parentheses and put a plus sign between them. We will get the following:

(− 3 x) + (2, 72 x 3 y 5 z)

When we do the parenthesis expansion, we get - 3 x + 2, 72 x 3 y 5 z. This is a polynomial, written in standard form, which will be the result of adding these monomials.

Answer:(− 3 x) + (2.72 x 3 y 5 z) = − 3 x + 2.72 x 3 y 5 z.

If we have three, four or more terms, we carry out this action in exactly the same way.

Example 2

Condition: carry out the indicated operations with polynomials in the correct order

3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

Solution

Let's start by opening the brackets.

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

We see that the resulting expression can be simplified by adding similar terms:

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c = = (3 a 2 + a 2 - 7 a 2) + 4 a c - 2 2 3 a c + 4 9 = = - 3 a 2 + 1 1 3 a c + 4 9

We have a polynomial, which will be the result of this action.

Answer: 3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c = - 3 a 2 + 1 1 3 a c + 4 9

In principle, we can add and subtract two monomials, subject to some restrictions, so that we end up with a monomial. To do this, you need to meet some conditions regarding addends and subtracted monomials. We will tell you how this is done in a separate article.

Rules for multiplying monomials

The multiplication action does not impose any restrictions on the factors. The monomials being multiplied do not have to meet any additional conditions in order for the result to be a monomial.

To perform multiplication of monomials, you need to follow these steps:

Write down the piece correctly.
Expand the parentheses in the resulting expression.
If possible, group factors with the same variables and numeric factors separately.
Perform the necessary operations with numbers and apply the property of multiplication of powers with the same bases to the remaining factors.

Let's see how this is done in practice.

Example 3

Condition: multiply the monomials 2 x 4 y z and - 7 16 t 2 x 2 z 11.

Solution

Let's start by composing the work.

We open the brackets in it and get the following:

2 x 4 y z - 7 16 t 2 x 2 z 11

2 - 7 16 t 2 x 4 x 2 y z 3 z 11

All we have to do is multiply the numbers in the first brackets and apply the property of powers for the second. As a result, we get the following:

2 - 7 16 t 2 x 4 x 2 y z 3 z 11 = - 7 8 t 2 x 4 + 2 y z 3 + 11 = = - 7 8 t 2 x 6 y z 14

Answer: 2 x 4 y z - 7 16 t 2 x 2 z 11 = - 7 8 t 2 x 6 y z 14 .

If our condition contains three or more polynomials, we multiply them using exactly the same algorithm. We will consider the issue of multiplying monomials in more detail in a separate material.

Rules for raising a monomial to a power

We know that a power with a natural exponent is the product of a certain number of identical factors. Their number is indicated by the number in the indicator. According to this definition, raising a monomial to a power is equivalent to multiplying the specified number of identical monomials. Let's see how it's done.

Example 4

Condition: raise the monomial − 2 · a · b 4 to the power 3 .

Solution

We can replace exponentiation with multiplication of 3 monomials − 2 · a · b 4 . Let's write it down and get the desired answer:

(− 2 · a · b 4) 3 = (− 2 · a · b 4) · (− 2 · a · b 4) · (− 2 · a · b 4) = = ((− 2) · (− 2) · (− 2)) · (a · a · a) · (b 4 · b 4 · b 4) = − 8 · a 3 · b 12

Answer:(− 2 · a · b 4) 3 = − 8 · a 3 · b 12 .

But what if the degree has a large indicator? It is inconvenient to record a large number of factors. Then, to solve such a problem, we need to apply the properties of a degree, namely the property of a product degree and the property of a degree in a degree.

Let's solve the problem we presented above using the indicated method.

Example 5

Condition: raise − 2 · a · b 4 to the third power.

Solution

Knowing the power-to-degree property, we can proceed to an expression of the following form:

(− 2 · a · b 4) 3 = (− 2) 3 · a 3 · (b 4) 3 .

After this, we raise to the power - 2 and apply the property of powers to powers:

(− 2) 3 · (a) 3 · (b 4) 3 = − 8 · a 3 · b 4 · 3 = − 8 · a 3 · b 12 .

Answer:− 2 · a · b 4 = − 8 · a 3 · b 12 .

We also devoted a separate article to raising a monomial to a power.

Rules for dividing monomials

The last operation with monomials that we will examine in this material is dividing a monomial by a monomial. As a result, we should obtain a rational (algebraic) fraction (in some cases it is possible to obtain a monomial). Let us immediately clarify that division by zero monomial is not defined, since division by 0 is not defined.

To perform division, we need to write down the indicated monomials in the form of a fraction and reduce it, if possible.

Example 6

Condition: divide the monomial − 9 · x 4 · y 3 · z 7 by − 6 · p 3 · t 5 · x 2 · y 2 .

Solution

Let's start by writing monomials in fraction form.

9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2

This fraction can be reduced. After performing this action we get:

3 x 2 y z 7 2 p 3 t 5

Answer:- 9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2 = 3 x 2 y z 7 2 p 3 t 5 .

The conditions under which, as a result of dividing monomials, we obtain a monomial, are given in a separate article.

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How to multiply powers? Which powers can be multiplied and which cannot? How to multiply a number by a power?

In algebra, you can find a product of powers in two cases:

1) if the degrees have the same bases;

2) if the degrees have the same indicators.

When multiplying powers with the same bases, the base must be left the same, and the exponents must be added:

When multiplying degrees with the same indicators, the overall indicator can be taken out of brackets:

Let's look at how to multiply powers using specific examples.

The unit is not written in the exponent, but when multiplying powers, they take into account:

When multiplying, there can be any number of powers. It should be remembered that you don’t have to write the multiplication sign before the letter:

In expressions, exponentiation is done first.

If you need to multiply a number by a power, you should first perform the exponentiation, and only then the multiplication:

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Addition, subtraction, multiplication, and division of powers