We draw your attention to the fact that in this section the concept of degree is analyzed only with a natural exponent and zero. The concept and properties of degrees with rational indicators (with negative and fractional) will be considered in lessons for grade 8. So, let’s figure out what the power of a number is. To record the product of a number by itself, an abbreviation is used several times. The degree symbol is here copy now –

°

Instead of multiplying six identical factors, 6 · 6 · 6 · 6 · 6 · 6 write 6 ^{6} and say “four to the sixth power”.

4 · 4 · 4 · 4 · 4 · 4 = 4 ^{6}

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

- 6 –
**degree basis ;** ^{6}– exponent.

In general, a degree with a base “ a ” and an indicator “ n ” is written using the expression:

__Remember!__

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

The notation “ a ^{n} ” reads like this: “ and to the power of n ” or “the nth power of the number a ”.

The exception is recorded:

- a
^{2}– it can be pronounced as ” a-squared”; - a
^{3}– it can be pronounced as “a in the cube.”

Of course, the expressions above can also be read by definition of degree:

- a
^{2}– ” and in the 2nd degree”; - a
^{3}– ” and in the 3rd degrees.”

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

__Remember!__

The power of the number “ a ” with the exponent n = 1 is the number itself:

a ^{1} = a

Any number in the zero degrees is equal to one.

a ^{0} = 1

Zero to any natural degree is zero.

0 ^{n} = 0

The unit to any degree is 1.

1 ^{n} = 1

The expression 0 ^{0} ( **zero to zero degree** ) is considered meaningless.

- (−32)
^{0}= 1 - 0
^{253}= 0 - 1
^{4}= 1

When solving examples, you need to remember that exponentiation refers to finding a numerical or literal value after raising it to power.

Example. Raise to a power.

- 5
^{3}= 5 · 5 · 5 = 125 - 2.5
^{2}= 2.5 · 2.5 = 6.25

**Raising to the power of a negative number**

The base of the degree (the number that is raised to the power) can be any number – positive, negative or zero.

__Remember!__

When raising a positive number to a power, a positive number is obtained.

When raising zero to natural power, zero is obtained.

When raising a negative number to the power, the result can be either a positive number or a negative number. It depends on whether the exponent was an even or odd number.

Consider examples of raising to the power of negative numbers.

From the examples examined, it can be seen that if a negative number is raised to an odd degree, then a negative number is obtained. Since the product of the 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.

__Remember!__

A negative number raised to an even power is a positive number.

A negative number raised to an odd degree is a negative number.

The square of any number is a positive number or zero, that is:

a ^{2} ≥ 0 for any a .

- 2 · (−3)
^{2}= 2 · (−3) · (−3) = 2 · 9 = 18 - −5 · (−2)
^{3}= −5 · (−8) = 40

#### How to make a degree symbol (solving examples)

When solving examples for raising to a power, they often make mistakes, forgetting that the entries (−5) ^{4} and −5 ^{4} are different expressions. The results of raising the power of these expressions will be different.

Calculating (−3) ^{4} means finding the fourth power of a negative number.

(−3) ^{4} = (−3) · (−3) · (−3) · (−3) = 81

While finding “ −3 ^{4} ” means that the example needs to be solved in 2 steps:

- Raise the fourth power to a positive number 3.

3^{4}= 3 · 3 · 3 · 3 = 81 - Put a minus sign in front of the result (that is, perform a subtraction action).

−3^{4}= −81

Example. Calculate: −6 ^{2} – (−1) ^{4}

−6 ^{2} – (−1) ^{4} = −37

- 6
^{2}= 6 · 6 = 36 - −6
^{2}= −36 - (−2)
^{4}= (−2) · (−2) · (−2) · (−2) = 16 - – (- 1)
^{4}= −1 - −36 – 1 = −37

**The procedure in the examples with degrees**

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

__Remember!__

In expressions with degrees that do not contain brackets, they first **enter the degree**, then multiply and divide, and finally add and subtract.

If the expression contains brackets, then first, in the above order, perform actions in brackets, and then the remaining actions in the same order from left to right.

Example. Calculate:

To facilitate the solution of examples, it is useful to know and use __the degree table__, which you can download for free on our website.