Number - Structure and calculation: calculate with roots, and with integer and fractional indices (Higher tier)

Overview

Fractional indices are a compact way of writing roots and powers together. The expression 27^{2/3} does not mean multiply 27 by two-thirds: it means take the cube root of 27, then square the result. A negative index adds one more instruction: take the reciprocal. Once candidates separate these instructions, calculations that look difficult become short, reliable sequences.

This guide covers Pearson Edexcel GCSE Mathematics 1MA1, specification N7. Roots and integer indices underpin both tiers; calculating with fractional indices is Higher-only. Exact surd calculations connect to Higher content in N8. These skills also support algebraic simplification, standard form, rearranging formulae and problems involving area or volume. 1

Typical questions ask candidates to work out an exact value, express an answer as a single power, explain a mistake or show why a relationship holds. A correct answer must match the requested form: 2^3 is a single power, whereas 8 is its numerical value. The examples below make the reasoning visible so that credit can be given for a correct method, even when an arithmetic slip affects the final answer. Practise without a calculator first, then check your understanding rather than merely checking a decimal.

Key Concepts

1. Integer indices describe repeated multiplication

In a^n, a is the base and n is the index, also called the exponent. For a positive integer n, the index counts how many copies of the base are multiplied. Thus 3^4=3\times3\times3\times3=81, not 3\times4.

The multiplication law follows by counting factors: a^3\times a^2 contains five copies of a, so it equals a^5. Division cancels matching factors, giving a^5\div a^2=a^3 for $a
e0. A power of a power multiplies indices:(a^3)^2=a^6$, because there are two groups of three factors.

Examiner focus: first identify the operation between the powers. Multiplication means add indices; division means subtract; a power applied to a power means multiply. These laws do not apply to addition: 2^2+2^3=4+8=12, not 2^5.

Cover and recall: simplify 7^3\times7^2 and (7^3)^2. Explain why the answers have different indices.

2. Zero and negative indices extend the same laws

For a non-zero base, dividing a power by itself gives 1. The division law also gives a^k\div a^k=a^{k-k}=a^0. Therefore a^0=1. It is not 0, and this rule does not define 0^0 in GCSE calculations.

Continue the pattern 2^3=8, 2^2=4, 2^1=2, 2^0=1. Each step divides by 2, so 2^{-1}=1/2 and 2^{-2}=1/4. In general, a^{-n}=1/a^n for $a
e0$.

A negative index changes the position of a factor, not its sign. Consequently 5^{-2}=1/25, whereas -5^2=-25 and (-5)^2=25. Brackets determine whether a negative sign belongs to the base.

Why does this work? Multiplying a^n by a^{-n} gives a^0=1. The two values must therefore be reciprocals. For a positive base, every defined power remains positive.

3. Why a half index represents a square root

Let x\ge0 and put y=x^{1/2}. The multiplication law gives y\times y=x^{1/2+1/2}=x. Therefore y^2=x. The notation x^{1/2} denotes the non-negative, or principal, square root, so x^{1/2}=\sqrt{x}.

There is an important distinction: \sqrt{25}=5, but solving t^2=25 gives t=5 or t=-5. The radical symbol selects the principal root; the equation asks for every value whose square is 25. A complete explanation needs this sign convention, not just the observation that a value squares to x.

Similarly, a^{1/3}=\sqrt$3{a}for positivea. The cube root of 64 is 4 because4^3=64. Odd roots of negative numbers can be real, such as\sqrt3{-8}=-2, whereas\sqrt{-8}$ has no real value. Use positive bases for the general fractional-index laws in this guide; do not extend them blindly to negative bases.

4. Fractional indices: denominator root, numerator power

For a>0, a^{m/n}=(\sqrt[n]{a})^m, where m is an integer and n is a positive integer. The denominator n tells you which root to take; the numerator m tells you which power to apply.

Example: 27^{2/3}=(\sqrt$3{27})^2=3^2=9. Reading the fraction upside down would produce(\sqrt{27})^3$, which is a different expression. The fraction is an instruction about operations, not an amount by which the base is multiplied.

It is usually easier to find the root first. Squaring 27 first gives 729, whose cube root is still 9, but recognising the cube root of 27 keeps the numbers small. The two orders are mathematically equivalent for positive bases: root first is an efficiency choice, not the only valid method.

Another example: 32^{2/5}=(\sqrt[5]{32})^2=2^2=4. Recognising 32=2^5 also gives (2^5)^{2/5}=2^2. Reduce an exponent such as 2/4 to 1/2 when this makes the required root clearer.

Cover and recall: what does each part of 81^{3/4} instruct you to do? Find its exact value before checking the answer.

5. Negative fractional indices combine three instructions

For a>0 and positive integers m,n, write a^{-m/n}=1/a^{m/n}. Then calculate the root and power in the denominator.

Example: 16^{-3/2}=1/16^{3/2}=1/(\sqrt{16})^3=1/4^3=1/64. The denominator 2 means square root. The numerator 3 means cube. The negative sign means reciprocal. None of these instructions says make the answer negative.

Writing the reciprocal first is a useful way to avoid losing the minus sign. Alternatively, calculate 16^{3/2}=64 first, then take its reciprocal. Credit should be given for an equivalent correct route under the relevant mark scheme. Avoid taking a reciprocal twice.

A useful size check depends on the base. If a>1, then a^{-r}<1 for r>0. If 0<a<1, a negative power gives a value greater than 1: (1/4)^{-1/2}=1/(1/2)=2. Therefore “negative index means a small answer” is not a universal rule.

6. Fractions and decimals can be bases

If the base is a fraction, apply the root to its numerator and denominator. For example, (4/9)^{-1/2}=1/\sqrt{4/9}=1/(2/3)=3/2. The reciprocal of a fraction swaps its numerator and denominator; it does not change its sign.

For a less familiar calculation, (81/16)^{-3/4}=1/(3/2)^3=1/(27/8)=8/27. Here the fourth roots of 81 and 16 are 3 and 2. Showing the intermediate fraction makes the method easy to follow.

A terminating decimal may be easier as a fraction: 0.25^{-1/2}=(1/4)^{-1/2}=2. Candidates who multiply 0.25 by -1/2 have confused an index with a multiplier. Translate the notation before calculating.

7. Simplifying to a single power requires a common base

The laws combine indices only when the bases match. Rewrite 8=2^3 and 4=2^2 before simplifying 8^{2/3}\times4^{-1/2}:

(2^3)^{2/3}\times(2^2)^{-1/2}=2^2\times2^{-1}=2^1.

If the expression is then divided by 2^{-2}, subtract the denominator's index: 2^{1-(-2)}=2^3. The numerical value is 8. Subtracting a negative adds; it does not make the final index negative.

Examiner focus: if the command is “write as a single power of 2”, finish with 2^3. If it is “work out”, give 8. If both forms are requested, label both. Do not lose the requested base by ending with an equivalent but differently based power.

8. Roots and exact surd calculations

A surd is an irrational root written exactly, such as \sqrt{2}. It is not merely any expression containing a root symbol: \sqrt{16}=4 is rational. Surd manipulation is related Higher-tier N8 content. 1

For non-negative a,b, \sqrt{a}\sqrt{b}=\sqrt{ab}. Thus \sqrt8\times\sqrt2=\sqrt{16}=4. Alternatively, \sqrt8=2\sqrt2, so the product is 2(\sqrt2)^2=4.

Addition is different: \sqrt8+\sqrt2=2\sqrt2+\sqrt2=3\sqrt2, not \sqrt{10}. The product rule does not become a sum rule. When an exact answer is requested, keep a simplified fraction or surd rather than rounding to a decimal. Rationalising denominators is a separate N8 extension; it is not needed to evaluate the integer answers in the central examples here.

Mathematical Relationships

For the fractional-index relationships below assume a,b>0, p,q are rational indices, m is an integer and n is a positive integer. Ordinary integer-index laws also apply to non-zero real bases where defined. Every index or root relationship below is Must memorise, not supplied on Pearson's 2026 Higher formula sheet. 2

• Must memorise: a^p a^q=a^{p+q}. Same base, multiplication.
• Must memorise: a^p/a^q=a^{p-q}. Same base, division; the denominator must not be zero.
• Must memorise: (a^p)^q=a^{pq}. A power applied to a power.
• Must memorise: (ab)^p=a^p b^p and (a/b)^p=a^p/b^p. A power applies to every factor or to both parts of a quotient.
• Must memorise: a^0=1 and a^{-p}=1/a^p for $a
  e0$.
• Must memorise: a^{1/n}=\sqrt[n]{a} and a^{m/n}=(\sqrt[n]{a})^m=\sqrt[n]{a^m}.
• Must memorise: a^{-m/n}=1/(\sqrt[n]{a})^m when m>0.
• Must memorise: $\sqrt a\sqrt b=\sqrt{ab}$ for $a,b\ge0$; $\sqrt a/\sqrt b=\sqrt{a/b}$ for $a\ge0,b>0$; and $\sqrt{a^2}=|a|$ for real $a$.

The vertical bars in $|a|$ mean absolute value, the non-negative distance from zero. In particular, $\sqrt{(-3)^2}=3$, not $-3$. Zero cannot be used as a base with a negative index because that would require division by zero.

Practical Applications and Linked Skills

Area, volume and scale

For a square, Must memorise: A=s^2, so the positive side length is s=A^{1/2}. A square of area 49\text{ cm}^2 has side length 7 cm. For a cube, Must memorise: V=s^3, so s=V^{1/3}. A cube of volume 125\text{ cm}^3 has side length 5 cm. Taking a root reverses the power and changes the physical quantity back to length.

For similar solids, Must memorise: volume scale factor =k^3 and area scale factor =k^2, where k is the length scale factor. A volume scale factor of 27 gives k=27^{1/3}=3 and an area scale factor of 27^{2/3}=9. This gives fractional indices a concrete meaning.

For wider formula work, Given on formula sheet: A=\pi r^2 for a circle and V=\text{cross-sectional area}\times\text{length} for a prism. The index laws themselves are still not supplied. 2

Graphs, calculators and units

Graph plotting is not the central skill assessed here. As a useful algebra link, y=x^{1/2} is the square-root graph with real domain x\ge0: it passes through (0,0), (1,1), (4,2) and (9,3). A root graph does not mean divide every coordinate by 2.

Paper 1H does not permit a calculator; Papers 2H and 3H do. 1 On calculator papers, enter the entire fractional exponent in brackets, such as 16^{(-3/2)}, using the calculator's power function. Check that its display matches the intended expression. Do not round intermediate values in an exact calculation.

There are no special unit conversions in pure index calculations. In geometry, Must memorise: 1\text{ m}=100\text{ cm}, 1\text{ m}^2=10,000\text{ cm}^2 and 1\text{ m}^3=1,000,000\text{ cm}^3. Area and volume conversion factors must be squared or cubed. This topic has no required practical.

Exam-Focused Retrieval and Audio

Try the worked examples before reading their solutions. Then attempt the practice questions under a short time limit and explain each mistake using the correct rule. Repeat missed questions the next day and again several days later. Testing yourself from memory is more useful than repeatedly recognising an answer on the page.

The ten-minute female-tutor podcast follows a one-minute introduction, five minutes of core explanations, two minutes of exam advice, a one-minute recall quiz and a one-minute summary. Pause whenever asked to calculate, say the rule aloud and write your working before revealing the answer.

Assessment Notes and Sources

The real June 2023 Paper 1H Question 14(b) is used below with its published three-mark allocation. Other worked examples and all practice questions are original Edexcel-style questions with illustrative mark schemes, not Pearson-issued assessment materials. Pearson's M, A and B abbreviations refer to method, accuracy and unconditional accuracy marks respectively. Individual mark schemes decide exactly what is credited. 3