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Fractional and negative indices

Indices can be fractions or negative. Each kind has a clear meaning: roots and reciprocals.

Negative indices

a-n = 1an
5-2 = 125
(23)-1 = 32: a negative power flips the fraction.

Fractional indices

a1n is the nth root: 2512 = 5 and 813 = 2.
amn: take the root first, then the power. 823 = (∛8)2 = 22 = 4

Putting them together

16-34: root first: ⁴√16 = 2. Power: 23 = 8. Negative: 18.
Worked example

Evaluate 2723

  1. Cube root of 27 is 3.
  2. 32 = 9

Answer: 9

Key idea

Negative power: reciprocal. Fractional power: the denominator is the root, the numerator is the power. Do the root first to keep numbers small.

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