Learn4024Paper 2 · CalculatorNumberIndices and standard form

Number · Subtopic 5 of 8

Indices and Standard Form

Multiply powers of the same base by adding indices, divide by subtracting, and raise a power to a power by multiplying. $a^0=1$ , $a^{-n}=\dfrac1{a^n}$ , and $a^{\frac1n}$ is the $n$ th root. Standard form $A\times10^n$ ( $1\le A<10$ ) writes huge and tiny numbers compactly.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use the laws of indices.
Evaluate zero, negative and fractional indices.
Write numbers in and out of standard form.
Calculate with numbers in standard form, with and without a calculator.

Pause and think

$2^3=8$ , $2^2=4$ , $2^1=2$ . Each step divides by $2$ . Continue the pattern: what are $2^0$ and $2^{-1}$ ?

Section 2 of 14

Core idea

Start hereNumbers too big to write

The Sun is about 150 000 000 000 m away; a bacterium is 0.000 002 m long. Counting zeros is error-prone. Powers of ten, and the laws of indices, let us write and compute with such numbers easily.

The big idea

$a^m\times a^n=a^{m+n}$ , $a^m\div a^n=a^{m-n}$ , $(a^m)^n=a^{mn}$ ; so $a^0=1$ , $a^{-n}=\frac1{a^n}$ , $a^{1/n}=\sqrt[n]a$ . Standard form: $A\times10^n$ , $1\le A<10$ .

1. Laws

Where the laws come from

$a^3\times a^2=(a\cdot a\cdot a)(a\cdot a)=a^5$
Count the $a$ 's: powers add.
$\frac{a^5}{a^2}=a^3$
Cancel two $a$ 's: powers subtract.
$\frac{a^2}{a^2}=a^0=1$
Anything over itself.
$\frac{a^2}{a^5}=a^{-3}=\frac1{a^3}$
Three $a$ 's left on the bottom.

Worked exampleSimplify

$3x^2\times4x^5$ ; $(2a^3)^4$ ; $16^{3/4}$ ; $5^{-2}$ .

Goal: Simplest form.
Strategy: Numbers multiply; powers follow the laws.

$12x^7$
$3\times4$ ; $2+5$ .
$2^4a^{12}=16a^{12}$
Everything in the bracket to the power 4.
$(\sqrt[4]{16})^3=2^3=8$
Root, then power.
$\frac1{25}$
Negative: reciprocal.

Answer

12x^7

16a^{12}

8

\frac1{25}

✓ Check

(2a^3)^4

a=1

16

✓.

Common misconception $(2a^3)^4\neq2a^{12}$ The 2 must be raised to the power too.

2. Standard form

Powers of 10 on a number line (each step is ×10)

Positive powers: big numbers. Negative powers: small numbers (less than 1).

Worked exampleStandard form calculations

Write 0.000 34; compute $(3.2\times10^4)(5\times10^{-7})$ , $\frac{6\times10^8}{1.5\times10^3}$ , $3\times10^5+4\times10^4$ .

Goal: Standard form answers.
Strategy: Numbers and powers separately; adjust so $1\le A<10$ .

$3.4\times10^{-4}$
Decimal point moves 4 places.
$16\times10^{-3}=1.6\times10^{-2}$
$16=1.6\times10$ .
$4\times10^5$
$6\div1.5$ ; $8-3$ .
$300000+40000=3.4\times10^5$
Adding: convert to ordinary numbers (or match powers).

Answer

3.4\times10^{-4}

1.6\times10^{-2}

4\times10^5

3.4\times10^5

✓ Check

32000\times0.0000005=0.016

✓.

Pause and think

Solve $2^x=\frac18$ .

Summary

Laws give meaning to zero, negative, fractional powers.
Standard form: $1\le A<10$ .

Index: A power.
Standard form: $A\times10^n$ .

HistoryCounting grains of sand. In The Sand Reckoner (around

250

BC) Archimedes estimated how many grains of sand would fill the universe, about

10^{63}

in our notation. Greek numerals couldn't handle such numbers, so he invented his own system of powers.

Fun factGoogle is a misspelling. A "googol" is

10^{100}

, a name invented by a nine-year-old in 1938. The search company's founders meant to name it after the googol, but the domain was registered with a misspelling: Google.

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