The idea

Zero and negative indices

Why anything to the power zero is 1 (not 0), why a negative index means a reciprocal (not a negative number), and why a fraction raised to a negative power flips.

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Number · Indices

Zero and negative indices

Why anything to the power zero is 1 (not 0), why a negative index means a reciprocal (not a negative number), and why a fraction raised to a negative power flips.

Why it works

What the division law forces

2^0 = 0

" and "

4^{-2} = -16

" are two of the most confidently wrong answers in GCSE number. The real values surprise people because zero and negative indices aren't new rules to memorise — they're what the division law forces to be true if the pattern of powers is to keep working.

Why a^0 = 1, not 0

Divide equal powers:

\frac{2^3}{2^3} = 2^{3-3} = 2^0, \qquad \text{but also} \qquad \frac{8}{8} = 1.

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