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Exponents – Running Notes
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Exponents (Level 1)

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Laws of Exponents Part 2

5. Power of Quotient Law

When a fraction is raised to a power, both numerator and denominator get the power.

For example:

$\left(\frac{a}{b}\right)^3 = \left(\frac{a}{b}\right) \times \left(\frac{a}{b}\right) \times \left(\frac{a}{b}\right)$

$= \frac{a^3}{b^3}$

Therefore,

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

Examples:

$\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$

$\left(\frac{2}{x}\right)^2 = \frac{2^2}{x^2} = \frac{4}{x^2}$

$\left(\frac{x}{2}\right)^3 = \frac{x^3}{2^3} = \frac{x^3}{8}$

6. Zero Exponent Law

Observe the following pattern:

$2^3 = 8$

$2^2 = \frac{8}{2} = 4$

$2^1 = \frac{4}{2} = 2$

$2^0 = \frac{2}{2} = 1$

Therefore,

$a^0 = 1, \quad a \neq 0$

Examples:

$5^0 = 1$

$100^0 = 1$

$30^0 = 1$

Important note:

$0^0 \text{ is undefined}$

7. Negative Exponent Law

A negative exponent means reciprocal.

Observe:

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

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

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

Therefore,

$a^{-n} = \frac{1}{a^n}$

Also,

$a^{-1} = \frac{1}{a}$

Examples:

$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$

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

$5^{-1} = \frac{1}{5}$

$6^{-2} = \frac{1}{6^2} = \frac{1}{36}$

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