Free sample · National 5 Mathematics

A first look at surds

Some square roots are whole numbers. Others cannot be written exactly as a fraction. A surd lets us keep those roots exact.

Compare two square roots

The square root of $9$ is exactly $3$, because $3 \times 3 = 9$.

The square root of $10$ lies between $3$ and $4$. Its decimal expansion goes on forever without repeating. Writing $\sqrt{10}$ keeps the value exact; writing $3.16$ gives a rounded approximation.

Key idea

A surd is an irrational root. Square roots of positive integers that are not perfect squares are examples.

$$\sqrt{2},\quad \sqrt{3},\quad \sqrt{10}$$

A short worked example

To simplify $\sqrt{12}$, look for a perfect-square factor of $12$. Since $12 = 4 \times 3$, we can write:

$$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$

The result is still exact. We have taken the square root of the perfect-square factor outside the radical.

Try it yourself

Simplify $\sqrt{20}$ by finding a perfect-square factor.

Reveal the worked answer

$$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$

The perfect-square factor is $4$, whose square root is $2$.

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