Negative indices

Example

Simplify d^4 \div d^5.

Using index laws for division, subtract the powers.

d^4 \div d^5 = d^{4 - 5} = d^{-1}. This is an example of a negative index.

But d^4 \div d^5 also equals \frac{d \times d \times d \times d}{d \times d \times d \times d \times d}.

Cancelling common factors gives \frac{\cancel{d} \times \cancel{d} \times \cancel{d} \times \cancel{d}}{\cancel{d} \times \cancel{d} \times \cancel{d} \times \cancel{d} \times d}, which gives d^4 \div d^5 = \frac{1}{d}.

So d^{- 1} =  \frac{1}{d}.

The rule for negative indices is a^{-m} = \frac{1}{a^m}

Question
  1. Simplify p^{-2}
  2. Simplify 3^{-3}
  • a^{-m} = \frac{1}{a^m} so p^{-2} = \frac{1}{p^2}
  • a^{-m} = \frac{1}{a^m} so 3^{-3} = \frac{1}{3^3} = \frac{1}{27}