Surds Worked Solutions — GCSE Maths

Find the largest square factor of $8$

Look for the biggest perfect square dividing $8$; here $4 = 2^2$.

Split the surd

Use $\sqrt{ab} = \sqrt{a}\times\sqrt{b}$ to separate the square part.

Root the square part

$\sqrt{4}$ is exactly $2$.

Write the simplified surd

So $\sqrt{8} = 2\sqrt{2}$.

Find the largest square factor of $12$

Look for the biggest perfect square dividing $12$; here $4 = 2^2$.

Split the surd

Use $\sqrt{ab} = \sqrt{a}\times\sqrt{b}$ to separate the square part.

Root the square part

$\sqrt{4}$ is exactly $2$.

Write the simplified surd

So $\sqrt{12} = 2\sqrt{3}$.

Find the largest square factor of $18$

Look for the biggest perfect square dividing $18$; here $9 = 3^2$.

Split the surd

Use $\sqrt{ab} = \sqrt{a}\times\sqrt{b}$ to separate the square part.

Root the square part

$\sqrt{9}$ is exactly $3$.

Write the simplified surd

So $\sqrt{18} = 3\sqrt{2}$.

Find the largest square factor of $20$

Look for the biggest perfect square dividing $20$; here $4 = 2^2$.

Split the surd

Use $\sqrt{ab} = \sqrt{a}\times\sqrt{b}$ to separate the square part.

Root the square part

$\sqrt{4}$ is exactly $2$.

Write the simplified surd

So $\sqrt{20} = 2\sqrt{5}$.

Find the largest square factor of $50$

Look for the biggest perfect square dividing $50$; here $25 = 5^2$.

Split the surd

Use $\sqrt{ab} = \sqrt{a}\times\sqrt{b}$ to separate the square part.

Root the square part

$\sqrt{25}$ is exactly $5$.

Write the simplified surd

So $\sqrt{50} = 5\sqrt{2}$.