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Negative and Fractional Indices

The index laws you already know extend beautifully to negative and fractional powers. A negative index means "one over", and a fractional index means "a root". Once these two ideas click, you can rewrite roots and reciprocals as powers and use the same rules everywhere.

30 min Algebraic Expressions
Edexcel AS Level Maths: 1.4 Negative and Fractional IndicesCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Interpret a negative index as a reciprocal
  • Interpret a unit fractional index as a root
  • Evaluate powers of the form a^(m/n)
  • Rewrite roots and reciprocals using index notation
  • Combine negative and fractional indices with the index laws
1

Negative indices mean reciprocals

A negative index tells you to take the reciprocal\textbf{reciprocal} — that is, "one over" the positive-power version. The size of the index still controls the power; the minus sign just flips it to the denominator.

a−n=1ana^{-n} = \dfrac{1}{a^{n}}
A negative power moves the term to the bottom of a fraction (and vice versa).
Example 1Evaluate 2−32^{-3}
1Flip to a reciprocal: 2−3=1232^{-3} = \dfrac{1}{2^{3}}.
223=82^{3} = 8.
Answer
18\dfrac{1}{8}
Example 2Write 1x5\dfrac{1}{x^{5}} using a single index
1Anything on the bottom moves up with a negative index.
Answer
x−5x^{-5}

Tip — A negative index does not make the answer negative — it makes it a fraction. 2⁻³ = 1/8, not −8.

2

Unit fractional indices are roots

A power of 1n\tfrac{1}{n} means the nn-th root\textbf{root}. So a power of 12\tfrac{1}{2} is a square root and a power of 13\tfrac{1}{3} is a cube root.

a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}
The denominator of the fraction is the order of the root.
Example 1Evaluate 251225^{\frac{1}{2}}
1A power of 12\tfrac{1}{2} is a square root: 25\sqrt{25}.
Answer
55
Example 2Evaluate 271327^{\frac{1}{3}}
1A power of 13\tfrac{1}{3} is a cube root: 273\sqrt[3]{27}.
Answer
33
3

General fractional indices

For a fraction like mn\tfrac{m}{n}, the denominator is the root\textbf{denominator is the root} and the numerator is the power\textbf{numerator is the power}. Do the root first to keep the numbers small, then raise to the power.

amn=(an)m=amna^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m} = \sqrt[n]{a^{m}}
Bottom of the fraction = root; top of the fraction = power.
Example 1Evaluate 8238^{\frac{2}{3}}
1Denominator 33 ⟶ cube root first: 83=2\sqrt[3]{8} = 2.
2Numerator 22 ⟶ then square: 22=42^{2} = 4.
Answer
44
Example 2Evaluate 163416^{\frac{3}{4}}
1Fourth root: 164=2\sqrt[4]{16} = 2.
2Then cube: 23=82^{3} = 8.
Answer
88

Tip — Always take the root before applying the power — it keeps the arithmetic manageable.

4

Putting it together with negative fractions

A negative and\textbf{and} fractional index combines both ideas: take the reciprocal, then apply the fractional power. The same index laws still apply throughout.

Example 1Evaluate (14)−32\left(\dfrac{1}{4}\right)^{-\frac{3}{2}}
1The negative power flips the fraction: (14)−32=432\left(\dfrac{1}{4}\right)^{-\frac{3}{2}} = 4^{\frac{3}{2}}.
2Square root of 44 is 22; then cube: 23=82^{3} = 8.
Answer
88

Tip — Rewriting roots as fractional powers lets you use the multiply/divide index laws on surds and reciprocals too.

Formula recap

a−n=1ana^{-n} = \dfrac{1}{a^{n}}
Negative index ⟶ reciprocal.
a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}
Unit fraction ⟶ root.
amn=(an)ma^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}
Denominator roots, numerator powers.
a0=1a^{0} = 1
Still holds for any non-zero base.

Common mistakes to avoid

Treating a negative index as a negative answer: 3⁻² = −9.
It is a reciprocal: 3⁻² = 1/3² = 1/9.
Swapping the root and power in a^(m/n), e.g. 8^(2/3) = ³√(8²)... done as (8³)...
Denominator is the root, numerator is the power: 8^(2/3) = (³√8)² = 2² = 4.
Forgetting to flip the fraction with a negative power: (1/4)^(−1/2) = 1/2.
Flip first: (1/4)^(−1/2) = 4^(1/2) = 2.

Key takeaways

  • Negative index → reciprocal: a⁻ⁿ = 1/aⁿ.
  • Fractional index → root: a^(1/n) = ⁿ√a.
  • For a^(m/n): root by the denominator, power by the numerator (root first).
  • Negative fractional powers combine both: flip, then apply the root and power.

Test yourself

Answer a few questions, then review anything you missed.

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