Surds

What you will learn

• 4.1 nth roots, for example ³√8 = 2.
• 4.2 Rational and irrational numbers; place them on a number line.
• NF 4.3 Simple quadratic surds a√b: simplify, add, subtract, multiply, divide, and rationalise simple denominators. The official notes say complicated mixed operations are not required.

You need first: Number and algebra basics, Indices and scientific notation

Key terms

Standard steps

The two rules

• √a × √b = √(ab)
• √a ÷ √b = √(a/b)
• k√a + m√a = (k + m)√a: add or subtract like surds only.

But √a + √b is not √(a + b). √9 + √16 = 7, but √25 = 5.

Simplify √n

1. Find the largest square factor of n: 4, 9, 16, 25, 36, 49, …
2. √n = √(square × rest) = √square × √rest.
3. Check that the number left under the root has no square factor: √72 = √(36 × 2) = 6√2, and 2 has none.

Add or subtract

1. Simplify each surd first.
2. Add only like surds (same number under the root), like collecting like terms: 2√3 + 3√3 = 5√3.

Rationalise a denominator √b

Multiply top and bottom by √b, because √b × √b = b.

Worked examples

= √(4×3) + √(9×3)
= 2√3 + 3√3
= 5√3

= 6√3 / (√3 × √3)
= 6√3 / 3
= 2√3

√8 × √6  = √48 = √(16 × 3) = 4√3
√72 ÷ √2 = √36 = 6

Common mistakes

• √(a + b) = √a + √b. It doesn't.
• Not using the largest square factor: √48 = 2√12 is not finished; keep going to 4√3.
• Adding unlike surds: 2√3 + 3√2 cannot be combined.
• Rationalising and forgetting to multiply the numerator.
• Typing into the calculator and giving a decimal when the question says "in surd form".

How the DSE tests it

• Section B coordinate geometry and circle questions often ask for an exact distance in surd form.
• Exact trigonometric values: sin 60° = √3/2, tan 30° = 1/√3 = √3/3.
• Quadratic formula answers, such as 2 ± √5.
• Paper 2: simplify surd expressions.
• Foundation-only students can use the calculator for decimals, but should still recognise a surd answer in MC options.

Questions you might not ask in class

More Form 3 maths topics