Year 9 Maths

Surds & Indices

Master the index laws, learn to simplify surds, and rationalise denominators to write expressions in their simplest form.

Index Laws

An index (or exponent) tells us how many times a base is multiplied by itself. The index laws give us shortcuts for working with powers.

Multiplication Law

a^m × aⁿ = a^m+n

Division Law

a^m ÷ aⁿ = a^m−n

Power of a Power

(a^m)ⁿ = a^mn

Zero Index

a⁰ = 1 (a ≠ 0)

Negative Index

a⁻ⁿ = 1 / aⁿ

Fractional Index

a^1/n = ⁿ√a

Key Point: Index laws only apply when the bases are the same. You cannot combine 2³ × 3² using these laws.

A surd is a root that cannot be simplified to a whole number, such as √2, √3, or √5. We simplify surds by finding perfect square factors.

Simplifying Process

√(a × b) = √a × √b

Find the largest perfect square factor, then simplify.

Step 1: Find perfect square factors. 72 = 36 × 2

Step 2: √72 = √(36 × 2) = √36 × √2 = 6√2

Step 1: 3√50 = 3√(25×2) = 3 × 5√2 = 15√2

Step 2: 2√18 = 2√(9×2) = 2 × 3√2 = 6√2

Step 3: 15√2 + 6√2 = 21√2

Rationalising means removing the surd from the denominator. Multiply both the numerator and denominator by the surd in the denominator.

Rationalising Formula

a √b = a × √b √b × √b = a√b b

Step 1: Multiply top and bottom by √3: (6 × √3) / (√3 × √3)

Step 2: = 6√3/3 = 2√3

●Index laws let you simplify expressions with the same base by adding, subtracting, or multiplying indices.
●A surd is an irrational root. Simplify by extracting perfect square factors.
●Like surds (same number under the root) can be added or subtracted.
●Rationalise by multiplying numerator and denominator by the surd in the denominator.
●Fractional indices: a^m/n means take the nth root, then raise to power m.