Surds and fractional indices · Higher notes

A surd keeps an irrational value exact instead of replacing it with a rounded decimal. Look for a square factor inside a square root: this lets you separate the whole-number part. Only like surds can be collected. Fractional indices connect powers to roots: the denominator tells you the root and the numerator tells you the power. A negative index means reciprocal, not a negative answer. Work with positive bases here so fractional powers have a clear real value.

Learning objectives

Simplify and combine surds while keeping exact values.
Evaluate fractional and negative indices.

Key ideas

For positive bases, a fractional index means root, then power.

a^{m/n}=\left(\sqrt[n]{a}\right)^m\quad(a>0;\ m\in\mathbb{Z},\ n\in\mathbb{Z}_{>0})

For a non-zero base and positive integer power, a negative index gives a reciprocal.

a^{-k}=\frac{1}{a^k}\quad(a\ne0,\ k\in\mathbb{Z}_{>0})

Take square factors outside the root. Collect only like surds.

\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)

Rationalise a single surd denominator by multiplying both parts by that surd.

\frac{1}{\sqrt a}=\frac{\sqrt a}{a}\quad(a>0)

For a two-term surd denominator, multiply both parts by its non-zero conjugate.

Worked examples

Example 1

Simplify the surd expression exactly.

\sqrt{50}+3\sqrt8

Extract the square factors.

\sqrt{50}=5\sqrt2,\qquad\sqrt8=2\sqrt2

5\sqrt2+3(2\sqrt2)=(5+6)\sqrt2

Answer

11\sqrt2

Example 2

Evaluate the fractional power.

27^{2/3}=\left(\sqrt[3]{27}\right)^2=3^2

Answer

9

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Simplify surds, rationalise a denominator or evaluate fractional indices.

AO2 · Reasoning

Show that two exact expressions are equivalent using valid algebra.

AO3 · Problem solving

Use exact surds in a length or area problem before simplifying the result.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

A root of a sum is not usually the sum of the roots. A negative index does not make the answer negative.

Try it yourself

Attempt each question before opening its answer.

Check 1: Evaluate 27^(2/3).

Answer: 9

The cube root of 27 is 3, then 3² = 9.

Check 2: Rationalise 6/√3.

Answer: 2√3

Multiply top and bottom by √3: 6√3/3 = 2√3.

Explain it in your own words

Explain why √72 simplifies to 6√2.
Write what the denominator and numerator of a fractional index mean.
Show why a negative index produces a reciprocal.

Keep practising

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Specification source

AQA Number specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.