Rukasha Maths

GCSE maths Beta

Powers, roots and standard form · Foundation notes

Foundation · Number · AQA references N6, N7, N9

A power is repeated multiplication, not multiplication by the index: 3⁴ means four factors of 3. A square root reverses squaring, and the √ symbol gives the non-negative root. Powers of 10 describe place value, which makes very large and very small numbers easier to write. In standard form, one number at least 1 and less than 10 is multiplied by an integer power of 10. Positive powers make the value larger; negative powers make it smaller. When multiplying powers with the same base, count the total number of factors by adding their indices.

Learning objectives

  • Interpret powers and square roots.
  • Write and calculate numbers in standard form.

Key ideas

A positive integer power means repeated multiplication.

34=3×3×3×33^4=3\times3\times3\times3

The square-root symbol gives the non-negative root.

x≥0for x≥0\sqrt{x}\ge0\quad\text{for }x\ge0

For the same non-zero base, add indices when multiplying.

aman=am+n(a≠0, m,n∈Z)a^m a^n=a^{m+n}\quad(a\ne0,\ m,n\in\mathbb Z)

For a non-zero base and positive integer index, use these zero and negative index rules.

a0=1,a−n=1an(a≠0, n∈Z>0)a^0=1,\qquad a^{-n}=\frac1{a^n}\quad(a\ne0,\ n\in\mathbb Z_{>0})

For a positive number, standard form uses an integer power of ten.

A×10n,1≤A<10,n∈ZA\times10^n,\qquad1\le A<10,\quad n\in\mathbb Z

Worked examples

Example 1

Write the small number in standard form.

The first factor must be between one and ten.

0.00072=7.210 0000.00072=\frac{7.2}{10\,000}
110 000=10−4\frac1{10\,000}=10^{-4}

Answer

7.2×10−47.2\times10^{-4}

Example 2

Multiply the numbers in standard form.

(3×104)(2×103)=(3×2)104+3=6×107(3\times10^4)(2\times10^3)=(3\times2)10^{4+3}=6\times10^7

Answer

6×1076\times10^7

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

Evaluate powers and roots; convert to and calculate with standard form.

AO2 · Reasoning

Explain why a standard-form answer needs its leading number adjusted.

AO3 · Problem solving

Compare very large or very small quantities written in different forms.

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 negative index means a reciprocal, not a negative answer. Check the first factor when giving standard form.

Try it yourself

Attempt each question before opening its answer.

Check 1: Calculate 4³.

Answer: 64

4³ = 4 × 4 × 4 = 64.

Check 2: Calculate (3 × 10⁴) × (2 × 10³).

Answer: 6 × 10⁷

Multiply the first factors: 3 × 2 = 6. Add the indices of 10: 4 + 3 = 7.

Explain it in your own words

  • Explain the difference between 5² and 5 × 2.
  • Convert one large and one small number into standard form.
  • Show why 10⁻³ equals 0.001.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes

Specification source

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