Negative indices

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Example

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

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 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

Simplify \(p^{-2}\)
Simplify \(3^{-3}\)

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Laws of indices - AQANegative indices