What is the power of the number? Exposure to the power of the negative number? Procedure in the examples with degrees

Please note that this section deals with the notion of a degree only with a natural indicator and zero.

The concept and properties of degrees with rational exponents (with negative and fractional) will be discussed in the lessons for grade 8.

So, let's understand what is the power of the number. To write the product of the number itself on itself several times use the abbreviated notation. Thus, instead of the product of six identical factors 4 • 4 • 4 • 4 • 4 • 4, they write 4 ⁶ and say “four to the sixth degree”.

The expression 4 ⁶ is called the power of the number, where:

• 4 - the basis of the degree ;
• 6 - exponent .

In general, the degree with the base "a" and the index "n" is written using the expression:

The degree of the number "a" with a natural index "n", greater than 1, is the product of "n" equal factors, each of which is equal to the number "a".

The record a ⁿ reads like this: “but to the power of n” or “the nth power of the number a”.

The exceptions are records:

• a ² - it can be pronounced as "a squared";
• a ³ - it can be pronounced as “but in a cube”.

Of course, the expressions above can be read to determine the degree:

• a ² - “and in the second degree”;
• a ³ - "and in the third degree."

Special cases occur when the exponent is one or zero (n = 1; n = 0).

The expression 0 ⁰ ( zero to zero power ) is considered to be meaningless.

• (-32) ⁰ = 1
• 0 ²⁵³ = 0
• 1 ⁴ = 1

When solving examples, one must remember that exponentiation is called finding the value of a degree.

Example. Raise to degree.

• 5 ³ = 5 • 5 • 5 = 125
• 2.5 ² = 2.5 • 2.5 = 6.25
• (

  3

  four

  ) ⁴ =

  3

  four

  •

  3

  four

  •

  3

  four

  •

  3

  four

  =

  3 • 3 • 3 • 3

  4 • 4 • 4 • 4

  =

  81

  256

Negative number

The basis of a degree (a number that is raised to a power) can be any number — positive, negative, or zero.

When raising a negative number to a power, the result can be both a positive number and a negative number. It depends on whether the exponent is odd or odd.

Consider examples of raising to the power of negative numbers.

From the considered examples it is clear that if a negative number is raised to an odd power, then a negative number is obtained. Since the product of an odd number of negative factors is negative.

If a negative number is raised to an even power, then a positive number is obtained. Since the product of an even number of negative factors is positive.

• 2 • (- 3) ² = 2 • (- 3) • (- 3) = 2 • 9 = 18
• - 5 • (- 2) ³ = - 5 • (- 8) = 40

Note!

When solving examples of exponentiation, they often make mistakes, forgetting that the entries (- 5) ⁴ and -5 ⁴ are different expressions. The results of the exponentiation of these expressions will be different.

Calculate (- 5) ⁴ means to find the value of the fourth power of a negative number.

While finding -5 ⁴ means that the example needs to be solved in 2 steps:

1. Raise to the fourth power a positive number 5.
   5 ⁴ = 5 • 5 • 5 • 5 = 625
2. Put the minus sign before the result (that is, perform the subtraction action).
   -5 ⁴ = - 625

Example. Calculate: - 6 ² - (- 1) ⁴

1. 6 ² = 6 • 6 = 36
2. -6 ² = - 36
3. (- 1) ⁴ = (- 1) • (- 1) • (- 1) • (- 1) = 1
4. - (- 1) ⁴ = - 1
5. - 36 - 1 = - 37

The procedure in the examples with degrees

The calculation of the value is called exponentiation action. This is the action of the third step.

Example. Calculate: