• 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.

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."

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.

Cube roots and n^th roots

Cube roots behave differently from square roots. A cube root of 125 is a number that cubes to give 125 — and that number is just 5, because 5³ = 125. There's only one real cube root (not two), because cubing a negative number keeps it negative, so negatives and positives never "collide" on the same answer the way they do with squaring.

This generalises into n^th roots:

• If n is even (square root, 4th root, 6th root...) → there's a positive AND negative real root (for a positive starting number), and negative numbers have no real n^th root at all.
• If n is odd (cube root, 5th root, 7th root...) → there's exactly ONE real root, and it works for both positive and negative numbers.

Estimating roots (no calculator)

You can pin down a root between two whole numbers by "sandwiching" it between roots you already know. To estimate √20: you know √16 = 4 and √25 = 5, and since 20 sits between 16 and 25, √20 must sit between 4 and 5 (closer to 5, since 20 is closer to 25).

Reciprocals — the bridge to negative powers

The reciprocal of a number is simply 1 divided by that number. For fractions, there's a fast shortcut: flip the numerator and denominator. The reciprocal of 3/2 is 2/3.

Here's the crucial link to indices: a reciprocal can be written using a power of −1. So 5⁻¹ literally means "the reciprocal of 5," i.e. 1/5. This extends to any negative index: 5⁻² means the reciprocal of 5², which is 1/25.

Why these laws exist (don't just memorise — understand)

Every single index law below can be worked out from scratch just by expanding the powers into repeated multiplication. That's genuinely the best way to remember them — if you ever forget a rule mid-exam, expand a small example and the rule will reappear.

Golden rule: index laws only work when the bases are the same. You cannot combine 2³ × 5² using any index law, because 2 and 5 are different bases — there's no shortcut, you just have to calculate each separately (8 × 25 = 200).

The full table of laws

Handling different bases

Sometimes you'll be given an expression like 2⁵ × 4³, where the bases (2 and 4) look different — but 4 is secretly 2², so you can rewrite everything in terms of the same base before applying the laws.

Solving equations with indices

A really common exam question gives you an equation like 6¹⁰ × 6ˣ = 6² and asks you to find x. The trick: once both sides have the same base, the powers themselves must be equal — so you can just solve the resulting simple equation.

What is standard form, and why bother?

Try writing out the mass of the Earth in full: 5,970,000,000,000,000,000,000,000 kg. It's unwieldy, easy to miscount zeroes in, and painful to compare against other huge numbers. Standard form solves this by expressing any number as a small "nice" number multiplied by a power of 10 — compact, easy to compare, and easy to calculate with.

• n < 0 → number is between 0 and 1 (a small decimal)
• n = 0 → number is between 1 and 10
• n > 0 → number is 10 or bigger

Writing large numbers in standard form

The trick is to picture the decimal point sliding — you're counting how many places it moves to get "a" into the 1–10 range, and that count becomes your power of 10.

Writing small numbers in standard form

Same idea, but now you're dividing by 10 repeatedly instead of multiplying — which is why the power becomes negative.

On a calculator

Use brackets around each number, and the ×10ˣ button to enter the power. If your calculator spits out an answer that isn't in standard form (e.g. 243 × 10²⁰), you can fix it by rewriting the leading number in standard form and combining the powers using the index laws: 243 × 10²⁰ = (2.43 × 10²) × 10²⁰ = 2.43 × 10²².

Multiplying without a calculator

Multiplication and division are the "easy" operations for standard form, because the powers of 10 don't need to match first — you just deal with the ordinary numbers and the powers separately, then recombine.

Dividing without a calculator

Adding and subtracting — the tricky ones

Unlike multiplying and dividing, you cannot add/subtract standard form numbers by working with the powers separately — that's because addition and subtraction only make sense when you're combining "like" quantities. Just like you can't add 3 apples + 2 oranges and call it 5 apple-oranges, you can't add 10³ and 10² together directly — the powers of 10 must match first.

Method 1 — convert to ordinary numbers (best for manageable sizes)

Method 2 — align the powers of 10 (best for very large/small powers, where converting to ordinary numbers is impossible)

• I can explain what a base and an index/power are, and evaluate simple powers.
• I know why a⁰ = 1 and a¹ = a, not just that it's true.
• I can find both square roots of a number and know √ only gives the positive one.
• I understand why negative numbers have no real square root, but do have a real cube root.
• I can estimate a root by sandwiching it between two known integer roots.
• I can connect reciprocals to negative indices (a⁻ⁿ = 1/aⁿ).
• I can apply all the laws of indices confidently, including with algebra.
• I can rewrite an expression so two different bases become the same base.
• I can solve equations where I compare powers once bases match (e.g. finding x).
• I can evaluate fractional and negative fractional indices without a calculator.
• I can convert any large or small number into standard form correctly.
• I always check that my leading number 'a' is between 1 and 10.
• I can multiply and divide numbers in standard form without a calculator.
• I can add and subtract numbers in standard form, aligning the powers of 10 first.
• I know when to use "convert to ordinary numbers" vs "align powers" for adding/subtracting.

How these questions usually appear on exams

• "Find the value of x" equations where you compare powers on matching bases.
• "Without using a calculator, evaluate..." — testing pure index law fluency, including negative and fractional powers.
• Converting a real-world large/small number (distances, masses, populations) into standard form.
• Multiply/divide/add/subtract two numbers given in standard form, requiring the final answer back in standard form.
• "Show that..." proof-style questions using index laws to simplify an expression to a target form.