Rukasha Maths

GCSE maths Beta

Powers, roots and standard form · Revision guide

Foundation · Number · 1 page in the PDF

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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}
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
6×1076\times10^7

Watch out

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