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Number & Sets · SS 1

Negative and fractional indices

Turn negative and fractional powers into fractions and roots.

  • a−n = 1an
  • a1/n = ⁿ√a
  • am/n = (ⁿ√a)m

What the letters mean

a
the base
n
a power; in a1/n it is the root to take (2 = square root, 3 = cube root)
m
the power to raise the root to

Worked example

Evaluate 82/3.

  1. Step 1: Take the root first (the bottom of the fraction)

    ∛8 = 2

  2. Step 2: Then raise to the power (the top)

    22 = 4

Final answer: 4

Do the root first: the numbers stay small and easy.

See it

Play with this picture from the Laws of Indices lesson.

222

×

2222

↓ all together: 7 factors

222
2222

23 × 24 = 23 + 4 = 27 = 128

m (first index)

3

n (second index)

4