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Surds, indices and exact algebra
Lesson 8 of 75
Standard practice

Negative fractional indices

Objective. Interpret reciprocal roots and solve index equations.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Negative fractional indices combine two rules: a negative power means 'reciprocal' and a fractional power means 'root', so a^(−m/n) = 1/(ⁿ√a)ᵐ. Read the denominator of the fraction as the root to take, the numerator as the power to raise it to, and the negative sign as flipping the whole result into a fraction. Solving equations with these indices often means rewriting both sides with a common base before comparing exponents. For example, 8^(−2/3) = 1/(∛8)² = 1/2² = 1/4.

Worked example

Solve 27^(−x/3) = 1/9 for x.

1/9
Fraction bar showing 1 out of 9 parts shaded
  1. 1Write 27 and 9 as powers of 3: 27 = 3³, 9 = 3²
  2. 2Rewrite the equation: (3³)^(−x/3) = 3^(−2)
  3. 3Simplify the exponent on the left: 3^(−x) = 3^(−2)
  4. 4Equate exponents: −x = −2, so x = 2

Answer: x = 2

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Common mistake. A frequent mistake is applying the negative sign to the base instead of taking the reciprocal, turning 4^(−1/2) into −2 rather than 1/√4 = 1/2; the negative index never produces a negative number for a positive base.

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