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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 Video by Zeeshan Zamurred Algebraic Expressions

Edexcel AS Level Maths: 1.4 Negative and Fractional IndicesWatch the full walkthrough 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

Negative indices mean reciprocals

A negative index tells you to take the $reciprocal$ — 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} = \frac{1}{a ^{n}}

A negative power moves the term to the bottom of a fraction (and vice versa).

1Flip to a reciprocal:

2^{- 3} = \frac{1}{2 ^{3}}

2^{3} = 8

Answer

\frac{1}{8}

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

Unit fractional indices are roots

A power of $\frac{1}{n}$ means the $n$ -th $root$ . So a power of $\frac{1}{2}$ is a square root and a power of $\frac{1}{3}$ is a cube root.

a^{\frac{1}{n}} = n a

The denominator of the fraction is the order of the root.

1A power of

\frac{1}{2}

is a square root:

25

Answer

5

General fractional indices

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

a^{\frac{m}{n}} = (n a)^{m} = n a^{m}

Bottom of the fraction = root; top of the fraction = power.

1Denominator

3

⟶ cube root first:

38 = 2

2Numerator

2

⟶ then square:

2^{2} = 4

Answer

4

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

Putting it together with negative fractions

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

1The negative power flips the fraction:

(\frac{1}{4})^{- \frac{3}{2}} = 4^{\frac{3}{2}}

2Square root of

4

2

; then cube:

2^{3} = 8

Answer

8

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

Formula recap

a^{- n} = \frac{1}{a ^{n}}

Negative index ⟶ reciprocal.

a^{\frac{1}{n}} = n a

Unit fraction ⟶ root.

a^{\frac{m}{n}} = (n a)^{m}

Denominator roots, numerator powers.

a^{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.

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Negative and Fractional Indices

30 min Video by Zeeshan Zamurred Algebraic Expressions