We figured out what the degree of a number is in general. Now we need to understand how to correctly calculate it, i.e. raise numbers to powers. In this material, we will analyze the basic rules for calculating the degree in the case of an integer, natural, fractional, rational and irrational exponent. All definitions will be illustrated with examples.

The concept of exponentiation

Let's start with the formulation of basic definitions.

Definition 1

Exponentiation is the calculation of the value of the power of some number.

That is, the words "calculation of the value of the degree" and "exponentiation" mean the same thing. So, if the task is "Raise the number 0 , 5 to the fifth power", this should be understood as "calculate the value of the power (0 , 5) 5 .

Now we give the basic rules that must be followed in such calculations.

Recall what a power of a number with a natural exponent is. For a power with base a and exponent n, this will be the product of the nth number of factors, each of which is equal to a. This can be written like this:

To calculate the value of the degree, you need to perform the operation of multiplication, that is, multiply the bases of the degree the specified number of times. The very concept of a degree with a natural indicator is based on the ability to quickly multiply. Let's give examples.

Example 1

Condition: Raise - 2 to the power of 4 .

Solution

Using the definition above, we write: (− 2) 4 = (− 2) (− 2) (− 2) (− 2) . Next, we just need to follow these steps and get 16 .

Let's take a more complicated example.

Example 2

Calculate the value 3 2 7 2

Solution

This entry can be rewritten as 3 2 7 · 3 2 7 . Earlier we looked at how to correctly multiply the mixed numbers mentioned in the condition.

Perform these steps and get the answer: 3 2 7 3 2 7 = 23 7 23 7 = 529 49 = 10 39 49

If the task indicates the need to raise irrational numbers to a natural power, we will need to first round their bases to a digit that will allow us to get an answer of the desired accuracy. Let's take an example.

Example 3

Perform the squaring of the number π .

Solution

Let's round it up to hundredths first. Then π 2 ≈ (3, 14) 2 = 9, 8596. If π ≈ 3 . 14159, then we will get a more accurate result: π 2 ≈ (3, 14159) 2 = 9, 8695877281.

Note that the need to calculate the powers of irrational numbers in practice arises relatively rarely. We can then write the answer as the power itself (ln 6) 3 or convert if possible: 5 7 = 125 5 .

Separately, it should be indicated what the first power of a number is. Here you can just remember that any number raised to the first power will remain itself:

This is clear from the record. .

It does not depend on the basis of the degree.

Example 4

So, (− 9) 1 = − 9 , and 7 3 raised to the first power remains equal to 7 3 .

For convenience, we will analyze three cases separately: if the exponent is a positive integer, if it is zero, and if it is a negative integer.

In the first case, this is the same as raising to a natural power: after all, positive integers belong to the set of natural numbers. We have already described how to work with such degrees above.

Now let's see how to properly raise to the zero power. With a base that is non-zero, this calculation always produces an output of 1 . We have previously explained that the 0th power of a can be defined for any real number not equal to 0 , and a 0 = 1 .

Example 5

5 0 = 1 , (- 2 , 56) 0 = 1 2 3 0 = 1

0 0 - not defined.

We are left with only the case of a degree with a negative integer exponent. We have already discussed that such degrees can be written as a fraction 1 a z, where a is any number, and z is a negative integer. We see that the denominator of this fraction is nothing more than an ordinary degree with a positive integer, and we have already learned how to calculate it. Let's give examples of tasks.

Example 6

Raise 2 to the -3 power.

Solution

Using the definition above, we write: 2 - 3 = 1 2 3

We calculate the denominator of this fraction and get 8: 2 3 \u003d 2 2 2 \u003d 8.

Then the answer is: 2 - 3 = 1 2 3 = 1 8

Example 7

Raise 1, 43 to the -2 power.

Solution

Reformulate: 1 , 43 - 2 = 1 (1 , 43) 2

We calculate the square in the denominator: 1.43 1.43. Decimals can be multiplied in this way:

As a result, we got (1, 43) - 2 = 1 (1, 43) 2 = 1 2 , 0449 . It remains for us to write this result in the form common fraction, for which it is necessary to multiply it by 10 thousand (see the material on the conversion of fractions).

Answer: (1, 43) - 2 = 10000 20449

A separate case is raising a number to the minus first power. The value of such a degree is equal to the number opposite to the original value of the base: a - 1 \u003d 1 a 1 \u003d 1 a.

Example 8

Example: 3 − 1 = 1 / 3

9 13 - 1 = 13 9 6 4 - 1 = 1 6 4 .

How to raise a number to a fractional power

To perform such an operation, we need to recall the basic definition of a degree with a fractional exponent: a m n \u003d a m n for any positive a, integer m and natural n.

Definition 2

Thus, the calculation of a fractional degree must be performed in two steps: raising to an integer power and finding the root of the nth degree.

We have the equality a m n = a m n , which, given the properties of the roots, is usually used to solve problems in the form a m n = a n m . This means that if we raise the number a to a fractional power m / n, then first we extract the root of the nth degree from a, then we raise the result to a power with an integer exponent m.

Let's illustrate with an example.

Example 9

Calculate 8 - 2 3 .

Solution

Method 1. According to the basic definition, we can represent this as: 8 - 2 3 \u003d 8 - 2 3

Now let's calculate the degree under the root and extract the third root from the result: 8 - 2 3 = 1 64 3 = 1 3 3 64 3 = 1 3 3 4 3 3 = 1 4

Method 2. Let's transform the basic equality: 8 - 2 3 \u003d 8 - 2 3 \u003d 8 3 - 2

After that, we extract the root 8 3 - 2 = 2 3 3 - 2 = 2 - 2 and square the result: 2 - 2 = 1 2 2 = 1 4

We see that the solutions are identical. You can use any way you like.

There are cases when the degree has an indicator expressed as a mixed number or decimal fraction. For ease of calculation, it is better to replace it with ordinary fraction and count as above.

Example 10

Raise 44.89 to the power of 2.5.

Solution

Let's convert the value of the indicator into an ordinary fraction: 44 , 89 2 , 5 = 44 , 89 5 2 .

And now we perform all the actions indicated above in order: 44 , 89 5 2 = 44 , 89 5 = 44 , 89 5 = 4489 100 5 = 4489 100 5 = 67 2 10 2 5 = 67 10 5 = = 1350125107 100000 = 13 501, 25107

Answer: 13501, 25107.

If the numerator and denominator of a fractional exponent are big numbers, then calculating such exponents with rational exponents is a rather difficult job. It usually requires computer technology.

Separately, we dwell on the degree with a zero base and a fractional exponent. An expression of the form 0 m n can be given the following meaning: if m n > 0, then 0 m n = 0 m n = 0 ; if m n< 0 нуль остается не определен. Таким образом, возведение нуля в дробную положительную степень приводит к нулю: 0 7 12 = 0 , 0 3 2 5 = 0 , 0 0 , 024 = 0 , а в целую отрицательную - значения не имеет: 0 - 4 3 .

How to raise a number to an irrational power

The need to calculate the value of the degree, in the indicator of which there is an irrational number, does not arise so often. In practice, the task is usually limited to calculating an approximate value (up to a certain number of decimal places). This is usually calculated on a computer due to the complexity of such calculations, so we will not dwell on this in detail, we will only indicate the main provisions.

If we need to calculate the value of the degree a with an irrational exponent a , then we take the decimal approximation of the exponent and count from it. The result will be an approximate answer. The more accurate the decimal approximation taken, the more accurate the answer. Let's show with an example:

Example 11

Calculate the approximate value of 2 to the power of 1.174367....

Solution

We restrict ourselves to the decimal approximation a n = 1 , 17 . Let's do the calculations using this number: 2 1 , 17 ≈ 2 , 250116 . If we take, for example, the approximation a n = 1 , 1743 , then the answer will be a little more precise: 2 1 , 174367 . . . ≈ 2 1 . 1743 ≈ 2 . 256833 .

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In continuation of the conversation about the degree of a number, it is logical to deal with finding the value of the degree. This process has been named exponentiation. In this article, we will just study how exponentiation is performed, while touching on all possible exponents - natural, integer, rational and irrational. And by tradition, we will consider in detail the solutions to examples of raising numbers to various degrees.

Page navigation.

What does "exponentiation" mean?

Let's start by explaining what is called exponentiation. Here is the relevant definition.

Definition.

Exponentiation is to find the value of the power of a number.

Thus, finding the value of the power of a with the exponent r and raising the number a to the power of r is the same thing. For example, if the task is “calculate the value of the power (0.5) 5”, then it can be reformulated as follows: “Raise the number 0.5 to the power of 5”.

Now you can go directly to the rules by which exponentiation is performed.

Raising a number to a natural power

In practice, equality based on is usually applied in the form . That is, when raising the number a to a fractional power m / n, the root of the nth degree from the number a is first extracted, after which the result is raised to an integer power m.

Consider solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We show two solutions.

First way. By definition of degree with a fractional exponent. We calculate the value of the degree under the sign of the root, after which we extract the cube root: .

The second way. By definition of a degree with a fractional exponent and on the basis of the properties of the roots, the equalities are true . Now extract the root Finally, we raise to an integer power .

Obviously, the obtained results of raising to a fractional power coincide.

Answer:

Note that the fractional exponent can be written as decimal fraction or a mixed number, in these cases it should be replaced by the corresponding ordinary fraction, after which exponentiation should be performed.

Example.

Calculate (44.89) 2.5 .

Solution.

We write the exponent in the form of an ordinary fraction (if necessary, see the article): . Now we perform raising to a fractional power:

Answer:

(44,89) 2,5 =13 501,25107 .

It should also be said that raising numbers to rational powers is a rather laborious process (especially when the numerator and denominator of the fractional exponent are quite large numbers), which is usually carried out using computer science.

In conclusion of this paragraph, we will dwell on the construction of the number zero to a fractional power. We gave the following meaning to the fractional degree of zero of the form: for we have , while zero to the power m/n is not defined. So, zero to a positive fractional power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions and 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the degree of a number with an irrational exponent. At the same time, in practical purposes it is usually enough to get the value of the degree up to some sign. We note right away that in practice this value is calculated using electronic computing technology, since manual raising to an irrational power requires a large number of cumbersome calculations. But nevertheless we will describe in general terms the essence of the actions.

To get an approximate value of the power of a with an irrational exponent, some decimal approximation of the exponent is taken, and the value of the exponent is calculated. This value is the approximate value of the degree of the number a with an irrational exponent. The more accurate the decimal approximation of the number is taken initially, the more accurate the degree value will be in the end.

As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of an irrational indicator: . Now let's raise 2 to a rational power of 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈ 2.250116. Thus, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of an irrational exponent, for example, , then we get a more accurate value of the original degree: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

We remind you that in this lesson we understand degree properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in lessons for grade 8.

An exponent with a natural exponent has several important properties that allow you to simplify calculations in exponent examples.

Property #1
Product of powers

Remember!

When multiplying powers with the same grounds the base remains the same, and the exponents are added.

a m a n \u003d a m + n, where " a"- any number, and" m", " n"- any natural numbers.

This property of powers also affects 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 as a degree.
  6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
• Present as a degree.
  (0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Important!

Please note that in the indicated property it was only about multiplying powers with the same grounds . 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 #2
Private degrees

Remember!

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

= 11 3 − 2 4 2 − 1 = 11 4 = 44

Example. Solve the equation. We use the property of partial degrees.
3 8: t = 3 4

T = 3 8 − 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 degree properties.
  = = = 2 9 + 2

  2 5

  = 2 11

  2 5

  = 2 11 − 5 = 2 6 = 64

  Important!

  Please note that property 2 dealt only with the division of powers with the same bases.

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

  Be careful!

  Property #3
  Exponentiation

  Remember!

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

  (a n) m \u003d a n m, where "a" is any number, and "m", "n" are any natural numbers.

  
  

  Properties 4
  Product degree

  Remember!

  When raising a product to a power, each of the factors is raised to a power. The results are then multiplied.

  (a b) n \u003d a n b n, where "a", "b" are any rational numbers; "n" - any natural number.

  • Example 1
    (6 a 2 b 3 c) 2 = 6 2 a 2 2 b 3 2 s 1 2 = 36 a 4 b 6 s 2
    
  • Example 2
    (−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6

  Important!

  Please note that property No. 4, like other properties of degrees, is also applied in reverse order.

  (a n b n)= (a b) n

  That is, to multiply degrees with the same exponents, you can multiply the bases, and leave the exponent unchanged.

  • Example. Calculate.
    2 4 5 4 = (2 5) 4 = 10 4 = 10,000
  • Example. Calculate.
    0.5 16 2 16 = (0.5 2) 16 = 1

  In more complex examples, there may be cases where multiplication and division must be performed on powers with different grounds and different indicators. In this case, we advise you to do the following.

  For example, 4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216
  

  Example of exponentiation of a decimal fraction.

  4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = 4

  Properties 5
  Power of the quotient (fractions)

  Remember!

  To raise a quotient to a power, you can raise the dividend and divisor separately to this power, and divide the first result by the second.

  (a: b) n \u003d a n: b n, where "a", "b" are any rational numbers, b ≠ 0, n is any natural number.

  • Example. Express the expression as partial powers.
    (5: 3) 12 = 5 12: 3 12

  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.

primary goal

To acquaint students with the properties of degrees with natural indicators and teach them to perform actions with degrees.

Topic “Degree and its properties” includes three questions:

• Determination of the degree with a natural indicator.
• Multiplication and division of powers.
• Exponentiation of product and degree.

Control questions

1. Formulate the definition of a degree with a natural exponent greater than 1. Give an example.
2. Formulate a definition of the degree with an indicator of 1. Give an example.
3. What is the order of operations when evaluating the value of an expression containing powers?
4. Formulate the main property of the degree. Give an example.
5. Formulate a rule for multiplying powers with the same base. Give an example.
6. Formulate a rule for dividing powers with the same bases. Give an example.
7. Formulate the rule for exponentiation of a product. Give an example. Prove the identity (ab) n = a n b n .
8. Formulate a rule for raising a degree to a power. Give an example. Prove the identity (a m) n = a m n .

Definition of degree.

degree of number a with a natural indicator n, greater than 1, is called the product of n factors, each of which is equal to A. degree of number A with exponent 1 the number itself is called A.

Degree with base A and indicator n is written like this: a n. It reads " A to the extent n”; “ n-th power of a number A ”.

By definition of degree:

a 4 = a a a a

. . . . . . . . . . . .

Finding the value of the degree is called exponentiation .

1. Examples of exponentiation:

3 3 = 3 3 3 = 27

0 4 = 0 0 0 0 = 0

(-5) 3 = (-5) (-5) (-5) = -125

25 ; 0,09 ;

25 = 5 2 ; 0,09 = (0,3) 2 ; .

27 ; 0,001 ; 8 .

27 = 3 3 ; 0,001 = (0,1) 3 ; 8 = 2 3 .

4. Find expression values:

a) 3 10 3 = 3 10 10 10 = 3 1000 = 3000

b) -2 4 + (-3) 2 = 7
2 4 = 16
(-3) 2 = 9
-16 + 9 = 7

Option 1

a) 0.3 0.3 0.3

c) b b b b b b b

d) (-x) (-x) (-x) (-x)

e) (ab) (ab) (ab)

2. Square the numbers:

3. Cube the numbers:

4. Find expression values:

c) -1 4 + (-2) 3

d) -4 3 + (-3) 2

e) 100 - 5 2 4

Multiplication of powers.

For any number a and arbitrary numbers m and n, the following is true:

a m a n = a m + n .

Proof:

rule : When multiplying powers with the same base, the bases remain the same, and the exponents are added.

a m a n a k = a m + n a k = a (m + n) + k = a m + n + k

a) x 5 x 4 = x 5 + 4 = x 9

b) y y 6 = y 1 y 6 = y 1 + 6 = y 7

c) b 2 b 5 b 4 \u003d b 2 + 5 + 4 \u003d b 11

d) 3 4 9 = 3 4 3 2 = 3 6

e) 0.01 0.1 3 = 0.1 2 0.1 3 = 0.1 5

a) 2 3 2 = 2 4 = 16

b) 3 2 3 5 = 3 7 = 2187

Option 1

1. Present as a degree:

a) x 3 x 4 e) x 2 x 3 x 4

b) a 6 a 2 g) 3 3 9

c) y 4 y h) 7 4 49

d) a a 8 i) 16 2 7

e) 2 3 2 4 j) 0.3 3 0.09

2. Present as a degree and find the value in the table:

a) 2 2 2 3 c) 8 2 5

b) 3 4 3 2 d) 27 243

Division of degrees.

For any number a0 and arbitrary natural numbers m and n such that m>n is true:

a m: a n = a m - n

Proof:

a m - n a n = a (m - n) + n = a m - n + n = a m

by definition of private:

a m: a n \u003d a m - n.

rule: When dividing powers with the same base, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.

Definition: The degree of a non-zero number with a zero exponent is equal to one:

because a n: a n = 1 for a0 .

a) x 4: x 2 \u003d x 4 - 2 \u003d x 2

b) y 8: y 3 = y 8 - 3 = y 5

c) a 7: a \u003d a 7: a 1 \u003d a 7 - 1 \u003d a 6

d) s 5:s 0 = s 5:1 = s 5

a) 5 7:5 5 = 5 2 = 25

b) 10 20:10 17 = 10 3 = 1000

V)

G)

e)

Option 1

1. Express the quotient as a power:

2. Find the values ​​of expressions:

Raising to the power of a product.

For any a and b and an arbitrary natural number n:

(ab) n = a n b n

Proof:

By definition of degree

(ab) n =

Grouping the factors a and factors b separately, we get:

=

The proved property of the degree of the product extends to the degree of the product of three or more factors.

For example:

(a b c) n = a n b n c n ;

(a b c d) n = a n b n c n d n .

rule: When raising a product to a power, each factor is raised to that power and the result is multiplied.

1. Raise to a power:

a) (a b) 4 = a 4 b 4

b) (2 x y) 3 \u003d 2 3 x 3 y 3 \u003d 8 x 3 y 3

c) (3 a) 4 = 3 4 a 4 = 81 a 4

d) (-5 y) 3 \u003d (-5) 3 y 3 \u003d -125 y 3

e) (-0.2 x y) 2 \u003d (-0.2) 2 x 2 y 2 \u003d 0.04 x 2 y 2

f) (-3 a b c) 4 = (-3) 4 a 4 b 4 c 4 = 81 a 4 b 4 c 4

2. Find the value of the expression:

a) (2 10) 4 = 2 4 10 4 = 16 1000 = 16000

b) (3 5 20) 2 = 3 2 100 2 = 9 10000= 90000

c) 2 4 5 4 = (2 5) 4 = 10 4 = 10000

d) 0.25 11 4 11 = (0.25 4) 11 = 1 11 = 1

e)

Option 1

1. Raise to a power:

b) (2 a c) 4

e) (-0.1 x y) 3

2. Find the value of the expression:

b) (5 7 20) 2

Exponentiation.

For any number a and arbitrary natural numbers m and n:

(a m) n = a m n

Proof:

By definition of degree

(a m) n =

Rule: When raising a power to a power, the base is left the same, and the exponents are multiplied.

1. Raise to a power:

(a 3) 2 = a 6 (x 5) 4 = x 20

(y 5) 2 = y 10 (b 3) 3 = b 9

2. Simplify expressions:

a) a 3 (a 2) 5 = a 3 a 10 = a 13

b) (b 3) 2 b 7 \u003d b 6 b 7 \u003d b 13

c) (x 3) 2 (x 2) 4 \u003d x 6 x 8 \u003d x 14

d) (y y 7) 3 = (y 8) 3 = y 24

A)

b)

Option 1

1. Raise to a power:

a) (a 4) 2 b) (x 4) 5

c) (y 3) 2 d) (b 4) 4

2. Simplify expressions:

a) a 4 (a 3) 2

b) (b 4) 3 b 5+

c) (x 2) 4 (x 4) 3

d) (y y 9) 2

3. Find the meaning of expressions:

Application

Definition of degree.

Option 2

1st Write the product in the form of a degree:

a) 0.4 0.4 0.4

c) a a a a a a a a a

d) (-y) (-y) (-y) (-y)

e) (bc) (bc) (bc)

2. Square the numbers:

3. Cube the numbers:

4. Find expression values:

c) -1 3 + (-2) 4

d) -6 2 + (-3) 2

e) 4 5 2 – 100

Option 3

1. Write the product as a degree:

a) 0.5 0.5 0.5

c) c c c c c c c c c

d) (-x) (-x) (-x) (-x)

e) (ab) (ab) (ab)

2. Present in the form of a square of the number: 100; 0.49; .

3. Cube the numbers:

4. Find expression values:

c) -1 5 + (-3) 2

d) -5 3 + (-4) 2

e) 5 4 2 - 100

Option 4

1. Write the product as a degree:

a) 0.7 0.7 0.7

c) x x x x x x

d) (-а) (-а) (-а)

e) (bc) (bc) (bc) (bc)

2. Square the numbers:

3. Cube the numbers:

4. Find expression values:

c) -1 4 + (-3) 3

d) -3 4 + (-5) 2

e) 100 - 3 2 5

Multiplication of powers.

Option 2

1. Present as a degree:

a) x 4 x 5 e) x 3 x 4 x 5

b) a 7 a 3 g) 2 3 4

c) y 5 y h) 4 3 16

d) a a 7 i) 4 2 5

e) 2 2 2 5 j) 0.2 3 0.04

2. Present as a degree and find the value in the table:

a) 3 2 3 3 c) 16 2 3

b) 2 4 2 5 d) 9 81

Option 3

1. Present as a degree:

a) a 3 a 5 e) y 2 y 4 y 6

b) x 4 x 7 g) 3 5 9

c) b 6 b h) 5 3 25

d) y 8 i) 49 7 4

e) 2 3 2 6 j) 0.3 4 0.27

2. Present as a degree and find the value in the table:

a) 3 3 3 4 c) 27 3 4

b) 2 4 2 6 d) 16 64

Option 4

1. Present as a degree:

a) a 6 a 2 e) x 4 x x 6

b) x 7 x 8 g) 3 4 27

c) y 6 y h) 4 3 16

d) x x 10 i) 36 6 3

e) 2 4 2 5 j) 0.2 2 0.008

2. Present as a degree and find the value in the table:

a) 2 6 2 3 c) 64 2 4

b) 3 5 3 2 d) 81 27

Division of degrees.

Option 2

1. Express the quotient as a power:

2. Find the values ​​of expressions:

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * ... * an = an.

For example, a=2, n=3: 2 * 2 * 2=2^3 = 8 .

In general, exponentiation is often used in various formulas in mathematics and physics. This function has a more scientific purpose than the four basic ones: Addition, Subtraction, Multiplication, Division.

Raising a number to a power

Raising a number to a power is not a difficult operation. It is related to multiplication like the relationship between multiplication and addition. Record an - a short record of the n-th number of numbers "a" multiplied by each other.

Consider exponentiation at the most simple examples moving on to complex ones.

For example, 42. 42 = 4 * 4 = 16 . Four squared (to the second power) equals sixteen. If you do not understand the multiplication 4 * 4, then read our article about multiplication.

Let's look at another example: 5^3. 5^3 = 5 * 5 * 5 = 25 * 5 = 125 . Five cubed (to the third power) equals one hundred and twenty-five.

Another example: 9^3. 9^3 = 9 * 9 * 9 = 81 * 9 = 729 . Nine cubed equals seven hundred twenty-nine.

Exponentiation Formulas

To correctly raise to a power, you need to remember and know the formulas below. There is nothing beyond natural in this, the main thing is to understand the essence and then they will not only be remembered, but also seem easy.

Raising a monomial to a power

What is a monomial? This is the product of numbers and variables in any quantity. For example, two is a monomial. And this article is about raising such monomials to a power.

Using exponentiation formulas, it will not be difficult to calculate the exponentiation of a monomial to a power.

For example, (3x^2y^3)^2= 3^2 * x^2 * 2 * y^(3 * 2) = 9x^4y^6; If you raise a monomial to a power, then each component of the monomial is raised to a power.

When raising a variable that already has a degree to a power, the degrees are multiplied. For example, (x^2)^3 = x^(2 * 3) = x^6 ;

Raising to a negative power

A negative exponent is the reciprocal of a number. What is a reciprocal? For any number X, the reciprocal is 1/X. That is X-1=1/X. This is the essence of the negative degree.

Consider the example (3Y)^-3:

(3Y)^-3 = 1/(27Y^3).

Why is that? Since there is a minus in the degree, we simply transfer this expression to the denominator, and then raise it to the third power. Just right?

Raising to a fractional power

Let's start the discussion on specific example. 43/2. What does power 3/2 mean? 3 - numerator, means raising a number (in this case 4) to a cube. The number 2 is the denominator, this is the extraction of the second root of the number (in this case 4).

Then we get the square root of 43 = 2^3 = 8 . Answer: 8.

So, the denominator of a fractional degree can be either 3 or 4, and to infinity any number, and this number determines the degree square root extracted from given number. Of course, the denominator cannot be zero.

Raising a root to a power

If the root is raised to a power equal to the power of the root itself, then the answer is the radical expression. For example, (√x)2 = x. And so in any case of equality of the degree of the root and the degree of raising the root.

If (√x)^4. Then (√x)^4=x^2. To check the solution, we translate the expression into an expression with fractional degree. Since the root is square, the denominator is 2. And if the root is raised to the fourth power, then the numerator is 4. We get 4/2=2. Answer: x = 2.

Anyway the best option just convert the expression to an expression with a fractional power. If the fraction is not reduced, then such an answer will be, provided that the root of the given number is not allocated.

Exponentiation of a complex number

What is a complex number? Complex number- an expression that has the formula a + b * i; a, b are real numbers. i is the number that, when squared, gives the number -1.

Consider an example. (2 + 3i)^2.

(2 + 3i)^2 = 22 +2 * 2 * 3i +(3i)^2 = 4+12i^-9=-5+12i.

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Exponentiation online

With the help of our calculator, you can calculate the exponentiation of a number to a power:

Exponentiation Grade 7

Raising to a power begins to pass schoolchildren only in the seventh grade.

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * … * an=an .

For example, a=2, n=3: 2 * 2 * 2 = 2^3 = 8.

Solution Examples:

Exponentiation presentation

Presentation on exponentiation, designed for seventh graders. The presentation may clarify some incomprehensible points, but there will probably not be such points thanks to our article.

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