2.2 Surds and rationalising the denominator — Practise with help

Work each one first, closed book. Before you reveal anything, write down the square numbers up to $144$ ; most simplifications need one of them.

Warm-up: remember

Write down the square numbers from $1^2$ to $12^2$ .
Expand $(x + 3)(x - 3)$ .
Simplify $\sqrt{4 \times 9}$ .

Answers

$1$ , $4$ , $9$ , $16$ , $25$ , $36$ , $49$ , $64$ , $81$ , $100$ , $121$ , $144$ .
$(x + 3)(x - 3) = x^2 - 9$ , a difference of two squares.
$\sqrt{4 \times 9} = \sqrt{36} = 6$ , and also $\sqrt{4}\sqrt{9} = 2 \times 3 = 6$ .

If question 2 was hard, revisit expanding brackets: the conjugate method in this section is the same expansion with a surd in place of the letter.

Practise with help

1. Simplify $\sqrt{48}$ .

Hint

Look for the largest square number that divides $48$ . The square numbers to test are $4$ , $9$ , $16$ , $25$ , $36$ .

Next step

$48$ is $16 \times 3$ , and $16$ is a square. Write $\sqrt{48} = \sqrt{16 \times 3}$ .

Full working

Split the square factor out of the root:

\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16}\,\sqrt{3} = 4\sqrt{3}.

2. Simplify $\sqrt{18} + \sqrt{50}$ .

Hint

Simplify each surd on its own first. The two simplified surds should become like terms.

Next step

Write $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ and do the same for $\sqrt{50}$ .

Full working

Simplify each surd:

\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}, \qquad \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}.

Both are multiples of $\sqrt{2}$ , so add the coefficients:

3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}.

3. Expand and simplify $\left(3 + \sqrt{2}\right)\left(1 - \sqrt{2}\right)$ .

Hint

Expand the brackets as usual, then use $\sqrt{2} \times \sqrt{2} = 2$ .

Next step

Multiply out to get $3 - 3\sqrt{2} + \sqrt{2} - 2$ , then collect the like terms.

Full working

Expand each pair of terms:

\left(3 + \sqrt{2}\right)\left(1 - \sqrt{2}\right) = 3 - 3\sqrt{2} + \sqrt{2} - \sqrt{2}\,\sqrt{2}.

Use $\sqrt{2}\,\sqrt{2} = 2$ :

3 - 3\sqrt{2} + \sqrt{2} - 2 = 1 - 2\sqrt{2}.

4. Write $\dfrac{8}{\sqrt{2}}$ with a rational denominator.

Hint

Multiply the top and the bottom by $\sqrt{2}$ , so the denominator becomes $\sqrt{2}\sqrt{2} = 2$ .

Next step

Work out $\dfrac{8\sqrt{2}}{2}$ .

Full working

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

\frac{8}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}.

5. Write $\dfrac{2}{5 - \sqrt{3}}$ with a rational denominator.

Hint

The conjugate of $5 - \sqrt{3}$ is $5 + \sqrt{3}$ . Multiply the top and the bottom by the conjugate.

Next step

The denominator becomes $5^2 - 3$ . Expand the numerator as $2\left(5 + \sqrt{3}\right)$ .

Full working

Multiply the top and the bottom by the conjugate:

\frac{2}{5 - \sqrt{3}} = \frac{2\left(5 + \sqrt{3}\right)}{\left(5 - \sqrt{3}\right)\left(5 + \sqrt{3}\right)}.

The denominator is a difference of two squares:

\left(5 - \sqrt{3}\right)\left(5 + \sqrt{3}\right) = 25 - 3 = 22.

\frac{2}{5 - \sqrt{3}} = \frac{2\left(5 + \sqrt{3}\right)}{22} = \frac{5 + \sqrt{3}}{11}.

6. A rectangle has length $\left(4 + \sqrt{5}\right)$ cm and width $\left(4 - \sqrt{5}\right)$ cm. Show that its area is a whole number, and work out the area.

Hint

The area is the product of the two lengths. The two factors are a conjugate pair, so the product is a difference of two squares.

Next step

Write the area as $\left(4 + \sqrt{5}\right)\left(4 - \sqrt{5}\right)$ and apply $x^2 - y^2$ with $x = 4$ and $y = \sqrt{5}$ .

Full working

The area is

\left(4 + \sqrt{5}\right)\left(4 - \sqrt{5}\right) = 4^2 - \left(\sqrt{5}\right)^2 = 16 - 5 = 11.

The area is exactly $11$ cm $^2$ , a whole number. The surds cancel because the lengths form a conjugate pair.