Cambridge (CIE) IGCSE — International Maths Extended
Powers, Roots & Standard Form
Everything you need to go from "I sort of remember the rules" to "I can prove any index law and convert any number to standard form without blinking."
💡 Big idea in one sentence: Powers are just repeated multiplication written in shorthand, roots undo that shorthand, and standard form uses powers of 10 to write really huge or really tiny numbers in a compact, comparable way.
SummaryThe whole chapter in one scan
- Powers (indices) tell you how many times to multiply a base by itself — e.g. 6³ = 6×6×6.
- Roots are the reverse operation — they ask "what number, raised to this power, gives me this result?"
- Square roots have two answers (±) when n is even; cube roots and other odd roots have one answer.
- Reciprocals (1 ÷ number) connect directly to negative indices: a⁻¹ = 1/a.
- The laws of indices are a toolkit of shortcuts for multiplying, dividing, and raising powers — including fractional powers, which are secretly roots in disguise.
- Standard form writes any number as
a × 10ⁿ, where 1 ≤ a < 10, making huge/tiny numbers easy to write, compare, and calculate with.
- To multiply/divide numbers in standard form: deal with the ordinary numbers and the powers of 10 separately, then tidy up.
- To add/subtract numbers in standard form: you must get the powers of 10 to match first — either by converting to ordinary numbers, or by rewriting one power to match the other.
1. Powers & RootsThe foundation everything else builds on
What are powers (indices)?
Think of a power as a lazy person's way of writing repeated multiplication. Instead of writing 6 × 6 × 6, we write 6³. The 3 is just counting how many 6's are being multiplied together.
The number being multiplied is called the base. The small floating number is called the index, power, or exponent — all three words mean the same thing, and examiners use them interchangeably, so don't get thrown off if a question suddenly says "exponent" instead of "power."
Two rules worth knowing by heart
a⁰ = 1 (for any non-zero a) | a¹ = a
In plain words: anything to the power of 0 collapses to 1, and anything to the power of 1 is just itself — the power hasn't "done" anything yet.
Why does a⁰ = 1? (the bit most students just memorise without understanding)
Look at the pattern: 2³=8, 2²=4, 2¹=2, 2⁰=? Each time the power drops by 1, you divide the previous answer by 2. So 2⁰ = 2 ÷ 2 = 1. This pattern holds for every base — which is exactly why the rule works.
Square roots
A square root of 25 is a number that, when squared, gives you 25 back. Here's the twist most students miss: both 5 and −5 square to give 25, because a negative times a negative is positive. So technically 25 has two square roots.
But the √ symbol itself is more polite than that — by convention, √25 only ever means the positive root, so √25 = 5, not ±5. If you want to show both roots, you write ±√25 = ±5 explicitly.
Negative numbers don't have a real square root — there's no real number you can square to get a negative result, since squaring always produces something ≥ 0.