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3. Powers with positive and negative exponents

The power operation is recorded as follows:

- base;

- exponent;

- the result of the increment or the value of the degree

reads: " a to the power b equals c "

Calculating a step value is called stepping!
If the magnifier is a natural number

\begin{array}{l} a^{n} = \underset{⏟}{a \cdot a \cdot a \cdot ... \cdot a} \\ n times \end{array}

The positive value of a positive integer is always positive.

3^{2} = 3 \cdot 3 = 9

{(\frac{2}{5})}^{3} = \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} = \frac{8}{125}

A negative number is positive if the multiplier is an even number and negative if the multiplier is an odd number.

{(- 3)}^{4} = (- 3) \cdot (- 3) \cdot (- 3) \cdot (- 3) = 81

{(- 3)}^{3} = (- 3) \cdot (- 3) \cdot (- 3) = - 27

Important!

Distinguish some seemingly similar notes

{(- 2)}^{4}

and

- 2^{4}

. In the first case, the minus sign is the base sign \((-2)\) and in the second case, the degree sign (base is \(2\)).

{(- 2)}^{4} = (- 2) \cdot (- 2) \cdot (- 2) \cdot (- 2) = 16

{- 2}^{4} = - (2 \cdot 2 \cdot 2 \cdot 2) = - 16

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