National 5 Maths worked question

Simplify and collect like surds

A fixed, reviewed surds question with a complete worked solution and method explanation.

Standard3 marksNon-calculator

Question

Simplify 3√5 + 2√20, giving your answer in exact form.

Show the worked solution

Worked solution

  1. √20 = √(4 × 5) = √4 × √5 = 2√5

    Use the square factor 4 to simplify √20.

  2. 2√20 = 2 × 2√5 = 4√5

  3. 3√5 + 4√5 = 7√5

    The terms are now like surds, so their coefficients can be added.

Final answer: 7√5

Why this method works

Surd terms can only be collected after their square roots have been simplified. Rewrite √20 as 2√5 first; both terms then contain √5.

Check your work

Square-factor checking confirms that (2√5)² = 20, so the simplification of √20 is correct

Common mistakes

  • Adding 5 and 20 inside one square root.
  • Writing √20 as 4√5 instead of 2√5
  • Converting the exact answer to a decimal.