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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9MA0Paper 2 · Pure Mathematics 2Algebra and functionsIndices and surds

Algebra and functions · Subtopic 6 of 16

Indices and Surds

$a^{\frac mn}$ means "take the $n$ th root, then raise to the power $m$ ", and $a^{-n}=\dfrac1{a^n}$ . Surds are roots left exact: simplify them by pulling out square factors, and remove them from denominators by multiplying by the conjugate.

Edexcel YMA01 WMA11Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2≈ 30 minRevision 6 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Use the laws of indices, including zero, negative and fractional powers.
Write expressions as a single power of $x$ .
Simplify surds such as $\sqrt{75}$ .
Rationalise denominators, including ones like $3-\sqrt3$ .

Pause and think

$x^2\times x^3=x^5$ because powers add. What must $x^{\frac12}\times x^{\frac12}$ be, and what does that tell you about $x^{\frac12}$ ?

Section 2 of 14

Core idea

Start hereWhat could 2 to the power ½ mean?

" $2^3$ " means three 2s multiplied. But "half a 2 multiplied"? Or "minus one 2s"? Nobody can multiply something half a time. Mathematicians gave these symbols meanings by insisting one rule keeps working: add the powers when you multiply. Everything else follows from that.

The big idea

$a^m\times a^n=a^{m+n}$ , $a^m\div a^n=a^{m-n}$ , $(a^m)^n=a^{mn}$ . Forced consequences: $a^0=1$ , $a^{-n}=\frac1{a^n}$ , $a^{1/n}=\sqrt[n]a$ , $a^{m/n}=(\sqrt[n]a)^m$ . Surds are irrational roots, simplified with $\sqrt{ab}=\sqrt a\sqrt b$ .

1. The laws and where the meanings come from

a^m\times a^n=a^{m+n}\qquad a^m\div a^n=a^{m-n}\qquad(a^m)^n=a^{mn}

$a$: the base (positive here).
$m,n$: the powers (indices).

Zero, negative and fractional powers

$a^3\div a^3=a^{0}\ \text{and}\ =1$
Anything divided by itself is 1, so $a^0=1$ .
$a^{-n}\times a^{n}=a^0=1\Rightarrow a^{-n}=\frac1{a^n}$
Adding powers; $a^{-n}$ must be the reciprocal.
$a^{1/2}\times a^{1/2}=a^1\Rightarrow a^{1/2}=\sqrt a$
The number that squares to $a$ .
$a^{m/n}=(a^{1/n})^m=(\sqrt[n]a)^m$
Power of a power.

y = 2ˣ: what negative and fractional powers mean

Each step left halves the value, so the pattern continues smoothly:

2^{-1}=\tfrac12

2^{-2}=\tfrac14

. Halfway between

2^0

and

2^1

2^{1/2}=\sqrt2

Worked exampleEvaluating

Evaluate $8^{2/3}$ , $27^{-1/3}$ and $(\tfrac{16}9)^{-3/2}$ .

Goal: Exact values.
Strategy: Negative: flip. Denominator: root. Numerator: power. Root first keeps numbers small.

$8^{2/3}=(\sqrt[3]8)^2=2^2=4$
Cube root then square.
$27^{-1/3}=\frac1{\sqrt[3]{27}}=\frac13$
Reciprocal of the cube root.
$\left(\tfrac{16}9\right)^{-3/2}=\left(\tfrac9{16}\right)^{3/2}=\left(\tfrac34\right)^3=\tfrac{27}{64}$
Flip, root, cube.

Answer

4

\frac13

\frac{27}{64}

✓ Check

4^{3/2}=8

, so

8^{2/3}=4

is the inverse ✓.

Worked exampleSolving $2^{x+1}=8^x$

Solve for $x$ .

Goal: $x$ .
Strategy: Write both sides as powers of 2, then equate powers.

$8^x=(2^3)^x=2^{3x}$
$8=2^3$ .
$2^{x+1}=2^{3x}\Rightarrow x+1=3x$
Equal powers of the same base.
$x=\tfrac12$
Solve.

Answer

x=\frac12

✓ Check

2^{1.5}=2.83

and

8^{0.5}=2.83

✓.

Common misconception $a^{-n}$ is not negative

2^{-3}=\frac18

, a positive number.

2. Surds

The diagonal of a unit square is √2

By Pythagoras the diagonal is

\sqrt{1^2+1^2}=\sqrt2

, a length that is real but cannot be written as a fraction.

\sqrt{ab}=\sqrt a\sqrt b\qquad\frac{1}{p-\sqrt q}=\frac{p+\sqrt q}{p^2-q}

$p+\sqrt q$: the conjugate: $(p-\sqrt q)(p+\sqrt q)=p^2-q$ has no surd.

Worked exampleSimplifying and rationalising

Simplify $\sqrt{50}+\sqrt{18}$ ; rationalise $\frac6{\sqrt3}$ and $\frac4{3-\sqrt5}$ .

Goal: Simplest forms.
Strategy: Extract square factors; multiply by $\sqrt3$ or the conjugate.

$\sqrt{25\cdot2}+\sqrt{9\cdot2}=5\sqrt2+3\sqrt2=8\sqrt2$
Like surds add.
$\frac{6}{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}=\frac{6\sqrt3}3=2\sqrt3$
$\sqrt3\sqrt3=3$ .
$\frac{4(3+\sqrt5)}{(3-\sqrt5)(3+\sqrt5)}=\frac{4(3+\sqrt5)}{4}=3+\sqrt5$
Difference of squares: $9-5=4$ .

Answer

8\sqrt2

;

2\sqrt3

;

3+\sqrt5

✓ Check

4/(3-2.236)=5.236=3+2.236

✓.

Common misconception $\sqrt{a+b}\neq\sqrt a+\sqrt b$

\sqrt{9+16}=5

, not

7

Pause and think

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

Summary

Three laws; meanings for $0$ , negative, fractional powers.
Simplify surds; rationalise with the conjugate.

Index: A power.
Surd: An irrational root such as √2.
Conjugate: $p+\sqrt q$ paired with $p-\sqrt q$ .

HistoryPowers that are fractions. The French scholar Nicole Oresme wrote about fractional powers in the 1300s, but the modern notation

a^{\frac mn}

and

a^{-n}

became standard only after Isaac Newton used it in the 1670s.

Fun factA dangerous number. Legend says the Pythagorean Hippasus proved that

\sqrt2

cannot be written as a fraction, and that his fellow Pythagoreans were so upset that he was drowned at sea. The story is probably exaggerated, but

\sqrt2

really is irrational.

Section 3 of 14

Key formulae

Laws of indices

a^m a^n=a^{m+n},\quad\frac{a^m}{a^n}=a^{m-n},\quad(a^m)^n=a^{mn}

Negative and fractional powers

a^{-n}=\frac1{a^n},\qquad a^{\frac mn}=\left(\sqrt[n]a\right)^m

Conjugates

(a+\sqrt b)(a-\sqrt b)=a^2-b

TrickRoot, power, flip. For something like

27^{-\frac23}

: the denominator

3

says cube root (

3

), the numerator

2

says square (

9

), and the minus sign says flip (

\tfrac19

). Do them in that order and the numbers stay small.

Section 4 of 14

Worked examples

Example 1 · Single power0 / 2 steps

Write

\dfrac{x^2\sqrt x}{x^{-1}}

as a single power of

x

Example 2 · Rationalising0 / 2 steps

Express

\dfrac{6}{3-\sqrt3}

in the form

a+b\sqrt3

Section 5 of 14

Explore the graph

Graphs of y = xᵏ

Loading the interactive graph…

Try this

With $k=0.5$ the curve is $y=\sqrt x$ . Every curve passes through $(1,1)$ . Why?
Make $k$ negative. What happens near $x=0$ , and why?
For which $k$ is the curve the straight line $y=x$ ?

Every power curve passes through $(1,1)$ because $1^k=1$ .

In the real worldPaper sizes. A-series paper (A4, A3, …) has sides in the ratio

1:\sqrt2

. Folding a sheet in half gives the next size down with exactly the same shape, which is only possible with that surd ratio.

Section 6 of 14

Common mistakes

Incorrect

8^{\frac23}=8\times\tfrac23

Correct

\left(\sqrt[3]8\right)^2=4

A fractional power is a root and a power, not a multiplication.

Incorrect

x^{-2}=-x^2

Correct

\dfrac1{x^2}

A negative power means reciprocal.

Incorrect

\sqrt{9+16}=3+4

Correct

\sqrt{25}=5

Roots split over multiplication, not addition.

IncorrectRationalising

3-\sqrt3

with

3-\sqrt3

CorrectUse

3+\sqrt3

Only the conjugate removes the surd.

$27^{-\frac23}$ looking terrifying

Peel it: cube root $3$ , square $9$ , flip: $\dfrac19$

Section 7 of 14

Exam tip

Exam tipShow the conjugate step. When rationalising, write the multiplication by

\dfrac{3+\sqrt3}{3+\sqrt3}

explicitly and expand the denominator to a whole number. That line is the method mark.

Section 8 of 14

Guided practice

SA-IAL-P1-AL-G012 marksL1 FoundationCalculatorIndices

Find the value of

8^{\frac23}

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-P1-AL1-0012 marksL1 FoundationNon-calculatorIndices

Find the value of

16^{\frac34}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL-0012 marksL1 FoundationCalculatorIndices

Find the value of

27^{-\frac23}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL-0022 marksL2 StandardCalculatorIndices

Write

\dfrac{x^2\sqrt x}{x^{-1}}

as a single power of

x

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL1-0023 marksL2 StandardNon-calculatorIndices

Solve

3^{2x-1}=27

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL-0042 marksL1 FoundationCalculatorSurds

Simplify

\sqrt{75}-\sqrt{12}

, giving your answer in the form

k\sqrt3

. State

k

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL1-0032 marksL1 FoundationNon-calculatorSurds

Simplify

\sqrt{48}+\sqrt{27}

, giving your answer in the form

k\sqrt3

. State

k

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL1-0042 marksL1 FoundationNon-calculatorSurds

Find the value of

(2+\sqrt3)(2-\sqrt3)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL1-0052 marksL1 FoundationNon-calculatorSurds

Write

\dfrac{10}{\sqrt5}

in the form

k\sqrt5

. State

k

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-AL-0033 marksL2 StandardCalculatorSurds

Express

\dfrac{6}{3-\sqrt3}

in the form

a+b\sqrt3

, where

a

and

b

are integers.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-IAL-P1-AL-0054 marksL3 ExamCalculatorSurds

Express

\dfrac{2+\sqrt5}{\sqrt5-1}

in the form

a+b\sqrt5

, where

a

and

b

are rational.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Add powers when multiplying, subtract when dividing, multiply for a power of a power.
$a^{-n}=\dfrac1{a^n}$ ; $a^{\frac mn}=(\sqrt[n]a)^m$ .
Simplify $\sqrt{ab}=\sqrt a\sqrt b$ using square factors.
Rationalise with the conjugate: $(a+\sqrt b)(a-\sqrt b)=a^2-b$ .

Section 12 of 14

Quick quiz

Q1 · SA-IAL-P1-AL-Q011 markL1 FoundationCalculatorIndices

a^{-\frac12}=

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WMA11 (Pure 1)
Where: P1 · 1.1–1.2 Indices and surds
Expected: Use the laws of indices for all rational exponents; use and manipulate surds, including rationalising the denominator.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: Short opening questions: "Simplify, giving your answer in the form a + b√3" or "Write as a single power of x".
Open in: YMA01 · WMA11 9MA0 · Paper 1 9MA0 · Paper 2

Section 14 of 14

Past paper questions

6 questions (39 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WMA11/01 · June 2025Question 15 marks
Write expressions with fractional and negative powers in the form kxⁿ.
Question paper, page 2 Mark scheme
WMA11/01 · June 2025Question 35 marks
Calculus with an expression in powers of x.
Question paper, page 6 Mark scheme
WMA11/01 · January 2025Question 49 marks
Work with indices, rewrite an equation by substitution and solve the quadratic.
Question paper, page 10 Mark scheme
WMA11/01 · June 2024Question 26 marks
Index laws and a surd equation, rationalising the answer.
Question paper, page 4 Mark scheme
WMA11/01 · June 2023Question 47 marks
Write an expression using indices, then differentiate.
Question paper, page 8 Mark scheme
WMA11/01 · June 2023Question 67 marks
Expand surds, rationalise a denominator and a proof.
Question paper, page 14 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9MA0, First teaching 2017 (official specification).