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Exponents – Running Notes
17

Exponents (Level 1)

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Laws of Exponents Part 3

8. Negative Power of a Fraction

For a fraction with a negative exponent, take the reciprocal and make the exponent positive.

Example:

$\left(\frac{2}{3}\right)^{-2} = \frac{1}{\left(\frac{2}{3}\right)^2}$

$= \frac{1}{\frac{2^2}{3^2}}$

$= \frac{3^2}{2^2}$

$= \left(\frac{3}{2}\right)^2$

Therefore,

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$

Examples:

$\left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3$

$= \frac{4^3}{3^3}$

$= \frac{64}{27}$

$\left(\frac{3}{2}\right)^{-2} = \left(\frac{2}{3}\right)^2$

$= \frac{4}{9}$

9. Powers of $-1$

When $-1$ is raised to an even power, the answer is $1$ . When $-1$ is raised to an odd power, the answer is $-1$ .

Examples:

$(-1)^2 = -1 \times -1 = 1$

$(-1)^3 = -1 \times -1 \times -1 = -1$

$(-1)^4 = -1 \times -1 \times -1 \times -1 = 1$

$(-1)^5 = -1 \times -1 \times -1 \times -1 \times -1 = -1$

Therefore,

$(-1)^{\text{even number}} = 1$

$(-1)^{\text{odd number}} = -1$

10. Powers of Negative Numbers

If a negative number is raised to an even power, the answer is positive. If a negative number is raised to an odd power, the answer is negative.

Examples:

$(-2)^2 = (-2) \times (-2) = 2^2 = 4$

$(-2)^3 = (-2) \times (-2) \times (-2) = -2^3 = -8$

Therefore,

$(-a)^n = a^n, \quad n \text{ is even}$

$(-a)^n = -a^n, \quad n \text{ is odd}$

Examples:

$(-3)^4 = 3^4 = 81$

$(-3)^5 = -3^5 = -343$

$(-5)^2 = 5^2 = 25$

$(-5)^3 = -5^3 = -125$

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