Powers, Roots & Standard Form (Cambridge IGCSE Maths): Revision notes

Section 1

What are indices and how do they work?

An index (or power) shows how many times a base is multiplied by itself. In $a^n$ , $a$ is the base and $n$ is the index.

Positive index: $a^n$ = a multiplied by itself n times
Zero index: $a^0 = 1$ (for any non-zero a)
Negative index: $a^{-n} = \frac{1}{a^n}$ (a reciprocal)
Fractional index (Extended): $a^{1/n} = \sqrt[n]{a}$ , and $a^{m/n} = (\sqrt[n]{a})^m$

Key termsindexbasereciprocal

Example

$16^{1/4} = \sqrt[4]{16} = 2$ , because 2 x 2 x 2 x 2 = 16.

Section 2

What are the rules of indices?

Rule	Formula
Multiplication	$a^m \times a^n = a^{m+n}$
Division	$a^m \div a^n = a^{m-n}$
Power of a power	$(a^m)^n = a^{mn}$

These rules only apply when the base is the same on both sides.

Key termsindex laws

Common mistake

A very common error is applying index rules to different bases, e.g. treating $2^3 \times 3^2$ as if the bases were equal — they are not, so the rules don't apply directly.

Section 3

What is standard form and how do I convert into and out of it?

Standard form writes numbers as $A \times 10^n$ , where $1 \leq A < 10$ and $n$ is a positive or negative integer.

Large numbers use a positive power: 5,600,000 = $5.6 \times 10^6$
Small numbers use a negative power: 0.00034 = $3.4 \times 10^{-4}$

Steps to convert into standard form:

Move the decimal point so only one non-zero digit remains before it
Count how many places you moved it — this is the power of 10
Moving left gives a positive power; moving right gives a negative power

Key termsstandard form

Section 4

How do I calculate with numbers in standard form?

Multiplying/dividing: multiply or divide the A values, then use the index laws on the powers of 10.

Example: $(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7$

Adding/subtracting: convert both numbers to the same power of 10 first (or convert to ordinary numbers), then combine.

Example: $5 \times 10^3 + 2 \times 10^2 = 5000 + 200 = 5200 = 5.2 \times 10^3$

Always check the final answer is still in correct standard form (A between 1 and 10) — adjust if needed.

Exam tip

After multiplying, if A is 10 or more, divide A by 10 and add 1 to the power to fix the standard form.

Must Know

$a^0 = 1$ ; $a^{-n} = 1/a^n$ ; $a^{1/n} = \sqrt[n]{a}$
Index laws: $a^m \times a^n = a^{m+n}$ , $a^m \div a^n = a^{m-n}$ , $(a^m)^n = a^{mn}$
Standard form: $A \times 10^n$ with $1 \leq A < 10$
Large numbers → positive power; small numbers → negative power
To add/subtract in standard form, match the powers of 10 first
Always leave the final answer with A between 1 and 10

That's the notes covered.

Carry on to the next subtopic.