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SQA National 5 Mathematics

Surds

Simplifying exact square-root expressions and rationalising simple denominators.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Know square numbers up to 144.
  • Remember that exact answers are often preferred to rounded decimals.
National 5 Mathematics lesson

Explanation

A surd is an irrational square-root expression left in exact form, such as √2 or 3√5.

At National 5, simplify surds by taking out square factors. For example, √72 = √(36 × 2) = 6√2.

When a denominator contains a single surd, multiply the top and bottom by that surd to rationalise it.

Visual support

Take out the square factor

√72 = √(36 × 2)

= √36√2

= 6√2

Key formulae and rules

  • √(ab) = √a√b
  • √(a²b) = a √b
  • 1 / √a = √a / a

Watch out

Writing √50 as √25 + √2

Check

Look for a square factor or matching base before reaching for a calculator.

Exam tip

For exact-form questions, leave the answer as a simplified surd and show the square factor used.

Calculator tip

Keep exact form until the final answer. Use decimals only when the question asks for them.

Worked examples

Worked example 1

Simplify a surd

Simplify √50

  1. Find a square factor: 50 = 25 × 2.
  2. Rewrite √50 as √(25 × 2)
  3. Take out √25 = 5

Answer: √50 = 5√2

Worked example 2

Collect like surds

Simplify 3√5 + 2√20

  1. Simplify √20 = √(4 × 5) = 2√5
  2. So 2√20 = 4√5
  3. Add like surds: 3√5 + 4√5

So: 7√5

Worked example 3

Rationalise a denominator

Write 6 / √3 with a rational denominator

  1. Multiply top and bottom by √3
  2. The denominator becomes √3 x √3 = 3.
  3. Simplify 6√3 / 3.

Answer: 2√3

Watch out

  • Writing √50 as √25 + √2
  • Combining unlike surds such as √2 + √3
  • Rounding when the question asks for exact form.