Simplifying Surds

Grade 7

What is a surd?

A surd is a square root (or other root) that cannot be simplified to a whole number or a fraction. It is irrational — the decimal goes on forever with no repeating pattern. We leave it in exact form using the √ symbol rather than rounding it.

For example, √9 = 3, so that is not a surd. But √5, √7, and √12 cannot be written exactly as decimals, so they are surds. Leaving the answer as √5 is more precise than writing 2.236…

Surds are exact. The moment you round a surd to a decimal, you lose accuracy. In GCSE questions that ask for an exact answer, a surd in its simplest form is what the mark scheme wants.

Simplifying by pulling out square factors

A surd is in its simplest form when the number inside the root has no square factors left. The method: find the largest square number that divides into the number under the root, split it out, then take the square root of that part.

√(a × b) = √a × √b

Use this rule to pull the square factor out front. The part that remains under the root is the bit that cannot be simplified further.

EXAMPLE
Simplify √12.
Find the largest square factor of 12. 4 × 3 = 12, and 4 is a perfect square. So: √12 = √(4 × 3) = √4 × √3 = 2 × √3 = 2√3. Check: (2√3)² = 4 × 3 = 12. ✓
EXAMPLE
Simplify √72.
The largest square factor of 72 is 36, because 36 × 2 = 72. So: √72 = √(36 × 2) = √36 × √2 = 6 × √2 = 6√2. If you spotted only 4 × 18 first, you get 2√18, and then need to simplify √18 again (since 9 × 2 = 18). It is quicker to find the largest square factor straight away.

Multiplying and dividing surds

Multiplying and dividing surds uses the same rule. Multiply (or divide) the numbers under the roots, then simplify if possible.

√a × √b = √(a × b) √a ÷ √b = √(a ÷ b)

You can also multiply a surd by a whole number in front: 3√5 × 2√3 = 6√15. Multiply the whole numbers together and the surds together.

EXAMPLE
Simplify √3 × √15.
√3 × √15 = √(3 × 15) = √45. Now simplify: the largest square factor of 45 is 9 (since 9 × 5 = 45). √45 = √(9 × 5) = 3√5. So the answer is 3√5.

Watch out

The most common mistake is picking a square factor that is not the largest one. For example, simplifying √72 as 2√18 is not wrong — but you have not finished. Always check whether the remaining number under the root still has a square factor hiding inside it.
Nova
Try this one

A square has an area of \( 20 \text{ cm}^2 \).

Find the exact length of one side of the square. Give your answer in the form \( a\sqrt{5} \) cm. [3]

Nova's hint:

This links the area of a square to surds. The side length of a square is the square root of its area.

Ask yourself: 'once I have a square root, can I simplify it using a square-number factor?' For example \( \sqrt{12} = 2\sqrt{3} \).

Try to Solve it with Nova?