Simplify \(d^4 \div d^5\).

\(d^4 \div d^5 = d^{4 - 5} = d^{-1}\). 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 \(a^{-m} = \frac{1}{a^m}\)

A negative power is often referred to as a reciprocal (\(a^{-m} = \frac{1}{a^m}\) is the reciprocal of \(a^m\)).

\(p^{-2} = \frac{1}{p^2}\)

\(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)

\(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\) (you may recognise this notation from standard form)