Fractional indices GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about fractional indices

A square with area 49 cm² has side length 7 cm, because 7 × 7 = 49. Finding that positive side length is taking the square root. The notation √49 means the square root of 49. We can write the same calculation as 49 raised to the power one half: a half power means a square root, not half the number.

See the idea first

The denominator names the root

An index is the small raised power in an expression. When we multiply powers of the same positive base, we add their indices. Two half powers therefore multiply to one whole power: the original number. So a half power is a square root; by convention, we choose the positive one. A one-third power similarly means a cube root. In a fractional index m/n, the denominator n names the root and the numerator m names the power afterwards.

Roots undo repeated multiplication
Known power	Undo with a root	Fractional index
$5^2=25$	$\sqrt{25}=5$	$25^{1/2}=5$
$3^3=27$	$\sqrt[3]{27}=3$	$27^{1/3}=3$
$2^4=16$	$\sqrt[4]{16}=2$	$16^{1/4}=2$

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a root

What the problem asks: Work out 64 raised to the power one third.

How to solve it: Find the number whose cube is 64. Since 4 × 4 × 4 = 64, the answer is 4.

Root then power

What the problem asks: Work out 27 raised to the power two thirds.

How to solve it: The denominator 3 means cube root: 27 becomes 3. The numerator 2 then means square: 3² = 9.

Use a negative fractional power

What the problem asks: Work out 16 raised to the power negative three quarters.

How to solve it: First take the fourth root of 16, giving 2, then cube it to get 8. The negative sign asks for its reciprocal, so the answer is 1/8, not −8.

A reliable routine

Evaluate a fractional power

Use this for a positive base raised to a rational power. Taking the root first keeps numbers smaller; the index laws show that root-then-power and power-then-root agree for positive bases.

Read the denominator of the index: choose that root.
Find the root of the base.
Raise the result to the numerator's positive power.
If the index is negative, take the reciprocal. Keep an exact fraction unless rounding is requested.

Check: √49 is 7. The equation x² = 49 has two solutions, 7 and −7: that is a different question. A fractional index is not an instruction to multiply the base by the fraction.

Fully worked

Fractional indices GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Half power

Higher only1 mark

Question

Evaluate $81^{1/2}$ .

81^{1/2}=\sqrt{81}=9

Because 9² = 81.

Example 2

Cube root

Higher only2 marks

Question

Evaluate $125^{1/3}$ .

125^{1/3}=\sqrt[3]{125}=5

Check: 5 × 5 × 5 = 125.

Example 3

Root then square

Higher only2 marks

Question

Evaluate $64^{2/3}$ .

The denominator tells us to take a cube root.

64^{2/3}=(\sqrt[3]{64})^2

=4^2=16

Example 4

Fourth root then cube

Higher only2 marks

Question

Evaluate $81^{3/4}$ .

81^{3/4}=(\sqrt[4]{81})^3

=3^3=27

Taking the root first avoids calculating 81³.

Example 5

Negative fractional power

Higher only3 marks

Question

Evaluate $32^{-2/5}$ .

32^{2/5}=(\sqrt[5]{32})^2=2^2=4

The negative index takes the reciprocal:

32^{-2/5}=\frac14

Example 6

Fraction base

Higher only3 marks

Question

Evaluate $\left(\frac9{16}\right)^{-1/2}$ .

Take the square root of numerator and denominator:

\left(\frac9{16}\right)^{1/2}=\frac34

Then take the reciprocal:

\left(\frac9{16}\right)^{-1/2}=\frac43

10 original questions · total 21 marks

Fractional indices GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 26 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Square root

Higher only1 mark

Evaluate $144^{1/2}$ .

Show worked answer

\sqrt{144}=12

2

Cube root

Higher only1 mark

Evaluate $216^{1/3}$ .

Show worked answer

\sqrt[3]{216}=6

Since 6³ = 216.

3

Fourth root

Higher only2 marks

Evaluate $16^{1/4}$ .

Show worked answer

\sqrt[4]{16}=2

Since 2⁴ = 16.

4

Two thirds

Higher only2 marks

Evaluate $8^{2/3}$ .

Show worked answer

8^{2/3}=(\sqrt[3]8)^2=2^2=4

5

Three halves

Higher only2 marks

Evaluate $36^{3/2}$ .

Show worked answer

36^{3/2}=(\sqrt{36})^3=6^3=216

6

Four fifths

Higher only2 marks

Evaluate $32^{4/5}$ .

Show worked answer

32^{4/5}=2^4=16

The fifth root of 32 is 2.

7

Negative half

Higher only2 marks

Evaluate $100^{-1/2}$ .

Show worked answer

100^{-1/2}=\frac1{\sqrt{100}}=\frac1{10}

8

Negative two thirds

Higher only3 marks

Evaluate $27^{-2/3}$ .

Show worked answer

27^{2/3}=3^2=9

27^{-2/3}=\frac19

9

Fraction base

Higher only3 marks

Evaluate $\left(\frac{16}{81}\right)^{3/4}$ .

Show worked answer

\sqrt[4]{\frac{16}{81}}=\frac23

\left(\frac23\right)^3=\frac8{27}

10

Find an index

Higher only3 marks

Find n if $64^n=16$ .

Show worked answer

Write both numbers as powers of 2:

2^{6n}=2^4

6n=4

n=\frac23

Check: cube root of 64 is 4, then square to get 16.

Examiner-style feedback

Common fractional indices mistakes

Half of the base

The half power of 36 is 6, not 18.

Swapping numerator and denominator

In 27 raised to the power two thirds, take the cube root then square, not the square root then cube.

Making the answer negative

A negative exponent means reciprocal. Positive bases still give positive answers.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Denominator: root.
Numerator: power.
Negative sign: reciprocal.
Check by rebuilding the original power.

Quick answers

Fractional indices FAQ

Can I calculate the power before the root?

For a positive base, yes. Root first usually keeps the arithmetic simpler.

Can I use a calculator?

When allowed, yes; use brackets around the entire fractional index. Many exam questions deliberately use exact powers you can recognise.

Build connected skills

What to revise next

Next skill

Laws of indices

Review integer powers and index laws.

Revise laws of indices

Next skill

Surds

Keep non-integer roots exact.

Revise surds

Content standards

Curriculum and rights review

N7 fractional indices are Higher tier. Examples use positive bases to avoid ambiguity in real-valued powers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content