Law 3: Negative Indices (Powers)

Topic Content:

This law applies when the power is negative.

\(\scriptsize a^{-n} = \normalsize \frac{1}{a^n}\)

In other words, an index number with a negative power is equal to the inverse of its index number with positive power. 

\(\scriptsize 3^{-3}= \normalsize \frac{1}{3^{3}}\)

\(\scriptsize 2^{-4}= \normalsize \frac{1}{2^{4}}\)

Worked Example 5.3.1:

Simplify the following:

a. \( \scriptsize 2^4 \: \div \: 2^8 \)

b. y²   ÷  y⁶

c. x⁵ ÷ x⁷

Solution

a. \( \scriptsize 2^4 \: \div \: 2^8 \\= \frac{\not{2} \: \times \: \not{2} \: \times \: \not{2}\: \times \: \not{2}}{\not{2} \: \times \: \not{2} \: \times \: \not{2}\: \times \: \not{2}\: \times \: 2\: \times \: 2\: \times \: 2 \: \times \: 2} \\= \frac{1}{2^4}\)

Alternatively, using