GCSE Maths Indices, surds & standard form revision

An index (power) counts repeated multiplication. The three core laws all follow from that counting: 2³ × 2⁴ = (2×2×2) × (2×2×2×2) = 2⁷, so multiplying powers with the same base adds indices. Dividing cancels factors, so 5⁶ ÷ 5⁴ = 5². Raising a power to a power repeats the whole block, so (3²)³ = 3⁶. The laws need the same base: 2³ × 3² cannot merge, and neither can 2³ + 2⁴ (index laws are about multiplication and division only).

The zero index completes the subtract rule: 7³ ÷ 7³ = 7⁰, but also equals 1, so 7⁰ = 1 — anything non-zero to the power zero is 1. Watch the brackets: −2² means −(2²) = −4, while (−2)² = 4. And a common trap: (2x)³ = 2³x³ = 8x³, not 2x³.

The subtract rule forces negative indices to mean reciprocals: 5² ÷ 5⁴ = 5⁻², but working it out gives 1⁄25, so 5⁻² = 1⁄5². In general a⁻ⁿ = 1⁄aⁿ, and the flip works both ways: 1⁄2⁻³ = 2³ = 8. Fractional indices mean roots: a^(1/2) = √a because squaring a^(1/2) gives a, and a^(1/3) = ∛a. The general power a^(m/n) = (√ⁿa)ᵐ — root first, then power, which keeps the numbers small: 8^(2/3) = (∛8)² = 2² = 4.

Negative and fractional combine: 27^(−1/3) = 1⁄∛27 = 1⁄3, and 16^(−3/4) = 1⁄(⁴√16)³ = 1⁄8. Evaluate the root before the power whenever you can — √16 = 4 then 4³ is friendlier than 16³ then a cube root.

A surd is an exact root that is not a whole number, like √2 or √5. The rule √(ab) = √a × √b lets you extract square factors: √12 = √4 × √3 = 2√3, √50 = 5√2, √72 = 6√2. Surds add only when they are the same surd: √27 + √12 = 3√3 + 2√3 = 5√3, just like 3x + 2x. Multiplying follows the same law: √8 × √6 = √48 = 4√3, and bracket expansions work as normal — (2 + √3)(2 − √3) = 4 − 3 = 1, because the surd terms cancel.

Rationalising removes a surd from a denominator. For a single surd, multiply top and bottom by it: 1⁄√3 = √3⁄(√3 × √3) = √3⁄3. For a bracket like 3 − √2, multiply by the conjugate: 1⁄(3 − √2) × (3 + √2)⁄(3 + √2) = (3 + √2)⁄(9 − 2) = (3 + √2)⁄7, since (a − √b)(a + √b) = a² − b. The answer must always end with a rational (surd-free) denominator.

Non-calculator papers ask for exact answers: leave π as π and surds as surds. A circle of radius 6 cm has circumference 2π × 6 = 12π cm — writing 37.7 cm is a rounded, not exact, answer. Sums mix shapes: √2 + √8 = √2 + 2√2 = 3√2, and ½ × √3 × 4 = 2√3. Fractions stay exact too: 2⁄7 + 1⁄3 = 13⁄21. When a calculator question says “give your answer as a multiple of π”, the π still stays in the final line.

Know which values are nice: (√2)² = 2, (2√5)² = 20, π² stays as π² (it is not 2π). Comparing sizes is fair game: π ≈ 3.14 so 3π ≈ 9.42, which sits between √81 = 9 and √144 = 12. Estimates come from rounding to one significant figure, but an “exact” answer and an “estimate” answer are different questions — read the command.

Standard form writes a number as A × 10ⁿ where 1 ≤ A < 10 and n is a whole number. Counting place values converts: 4 500 000 = 4.5 × 10⁶ (the point moved six places) and 0.00072 = 7.2 × 10⁻⁴ (four places the other way). Big numbers take positive indices; small numbers take negative ones. A number like 23 × 10⁴ is not standard form because 23 is not between 1 and 10 — rewrite it as 2.3 × 10⁵.

To multiply, multiply the A values and add the indices: (3 × 10⁵) × (4 × 10⁻²) = 12 × 10³ = 1.2 × 10⁴ (re-normalise because 12 is not allowed). To divide, divide the A values and subtract the indices: (8 × 10⁷) ÷ (2 × 10³) = 4 × 10⁴. For addition and subtraction, match the powers of ten first: 2.5 × 10⁶ + 3.7 × 10⁵ = 25 × 10⁵ + 3.7 × 10⁵ = 28.7 × 10⁵ = 2.87 × 10⁶. Keep Calculator papers honest: give final answers in standard form when the question says so, with the right number of significant figures when it does not.