A Student Guide to A-Level Mathematics

Dividing Expressions That Contains Exponents

You saw that multiplying adds the number of things being multiplied, increasing the exponent.

When you divide, the opposite happens:

\[a^m \div a^n = a^{m-n} \,.\]

This is because \(a^n\) cancels \(n\) lots of \(a\) from \(a^m\), so for example,

\[ \frac{2^6}{2^4} = \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2} = 2 \times 2 \times \frac{2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2} = 2^2\,.\]

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