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MathidiaMathematics Academy
Indices, fractional powers and surds
Lesson 6 of 75
Introduction

Fractional indices

Objective. Interpret roots as fractional powers and evaluate exact values.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A fractional index expresses a root as a power, so that a^(1/n) means the nth root of a, and a^(m/n) means the nth root of a raised to the power m. For example, 8^(1/3) = ∛8 = 2, and 8^(2/3) = (∛8)² = 2² = 4. The denominator of the fraction tells you which root to take, and the numerator tells you the power to apply, usually after taking the root to keep numbers small. Negative fractional indices combine this with the reciprocal rule, so a^(−m/n) = 1 ÷ a^(m/n).

Worked example

Evaluate 27^(2/3), writing your answer as an exact integer, using fractional indices.

2/3
Fraction bar showing 2 out of 3 parts shaded
  1. 1The denominator 3 means take the cube root of 27 first.
  2. 2∛27 = 3.
  3. 3The numerator 2 means square this result: 3² = 9.
  4. 4So 27^(2/3) = 9.

Answer: 9

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Common mistake. Pupils often compute 8^(2/3) by cubing first then finding a root of a huge number instead of rooting first. Always take the root (denominator) before applying the power (numerator) to keep the arithmetic simple.

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