GCSE Surds Practice Questions

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 $75$

Look for the biggest perfect square dividing $75$; 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 $\sqrt{75} = 5\sqrt{3}$.

Multiply top and bottom by $\sqrt{2}$

Clear the surd from the bottom.

Multiply the numerators

The top becomes $8\sqrt{2}$.

Multiply the denominators

The bottom becomes $2$.

Write the fraction

Now cancel the numbers.

Cancel

$8$ divided by $2$ is $4$.

State the answer

So $8/\sqrt{2} = 4\sqrt{2}$.

Set up the expansion (FOIL)

Multiply every term in the first bracket by every term in the second.

First product

A surd times itself gives $3$.

Outer product

Coefficient in front of the surd.

Inner product

Another surd term.

Last product

A whole number.

Write all four terms

Line up the four products.

Collect the whole numbers

Add the rational parts.

Collect the surds

Add the coefficients of $\sqrt{3}$.

State the answer

So the expansion is $11 + 6\sqrt{3}$.

Decimal check

Both the product and the answer equal about $21.392$.

Recognise difference of two squares

Here $a = 2$ and $b = \sqrt{3}$.

Square the first

$a^2 = 4$.

Square the second

$b^2 = 3$.

Subtract

So the product is $1$.

Check by full expansion

The $\sqrt{3}$ terms cancel.

Confirm cancellation

Leaving $4 - 3 = 1$.

Judge the claim

$1$ is a whole number, so it is rational.

Conclude Ravi is wrong

The product is $1$, which is rational, so Ravi is wrong.

Reason

Conjugate surds always multiply to a rational number.

Confirm

The exact value is $1$.

Underline the final answer

Write the answer on its own line so it is clear and easy to mark.

Check it answers the question

Re-reading the question, this is exactly what was required.

State the conclusion in one line

The working above leads directly to this result.

Match to the options

Compare each choice; only the option agreeing with this working is correct.

Sense check

The result is consistent with the decimal estimates found along the way.