National 5 Maths worked question

Simplify two surds before collecting

Practise simplifying surds and collecting like terms, then check each step with a full worked solution.

Standard3 marksNon-calculator

Question

Simplify 2√12 + √27, giving an exact answer.

Show the worked solution

Worked solution

  1. 2√12 = 2√(4 × 3) = 2 × 2√3 = 4√3

    Extract the square factor 4 from √12, then include the coefficient 2 outside the root

  2. √27 = √(9 × 3) = 3√3

    Extract the square factor 9 from √27

  3. 4√3 + 3√3 = 7√3

    Both terms now contain √3, so add their coefficients

Final answer: 7√3

Why this method works

Each surd must be simplified independently before terms can be collected. Extracting square factors turns both roots into multiples of √3, after which the coefficients behave like coefficients of an algebraic term.

Check your work

Check the component simplifications: (2√3)2 = 12 and (3√3)2 = 27. This confirms that √12 = 2√3 and √27 = 3√3 before collecting.

Common mistakes

  • Forgetting that the outside coefficient 2 multiplies the simplified value of √12
  • Adding 12 and 27 under one square root.
  • Converting the exact result to a decimal.