The idea

Surds and indices

The laws of indices — and why negative and fractional powers have to mean what they mean — plus how to simplify surds and rationalise a denominator.

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Pure · Algebra & functions

The laws of indices — and why negative and fractional powers have to mean what they mean — plus how to simplify surds and rationalise a denominator.

The three index laws are bookkeeping for repeated multiplication:

a^m \times a^n = a^{m+n}, \qquad a^m \div a^n = a^{m-n}, \qquad (a^m)^n = a^{mn}.

Count the factors and they're obvious:

a^2 \times a^3 = (a\,a)(a\,a\,a) = a^5

.

The interesting part is what happens when the power isn't a counting number. You don't get to choose what

a^0

a^{-n}

a^{1/2}

mean — they're forced, because the laws above have to keep working.

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The rest of the explanation, plus 3 worked examples you step through move by move.

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