AQA Level 2 Further Maths · Level 2 Certificate · Chapter 1: Number and algebra I · Lesson 4

Manipulating surds

About 75 minutesNo calculator63 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier lessons.

A1
Needed today
Write $72$ in the form $a b$ , where $a$ and $b$ are integers and $b$ is as small as possible.
[2 marks]
Answer
A2
Needed today
Expand and simplify $(2 x + 7) (2 x - 7)$ .
[2 marks]
Answer
A3
Last lesson
Work out the coefficient of $x^{2}$ in the expansion of $(1 - 2 x)^{5}$ .
[2 marks]
Answer
A4
From chapter 1
Solve $\frac{3 x + 1}{4} = \frac{x + 5}{2}$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Simplify $245 - 20 + 80$ .

Hint Write each root with the largest square factor taken out. Then they are all multiples of

5

B1
Simplify $312 + 75 - 227$ .
[3 marks]
Answer

Example 2

Expand and simplify $(3 - 25) (1 + 5)$ .

Hint Four products, as with

(a + b) (c + d)

. Remember

5 \times 5 = 5

B2
Expand and simplify $(7 - 2) (37 + 4)$ .
[2 marks]
Answer

Example 3

Rationalise the denominator of $\frac{5 + 3}{2 - 3}$ . Give your answer in the form $a + b 3$ .

Hint Multiply the top and the bottom by

2 + 3

, the bottom with its sign changed. The bottom becomes

4 - 3

B3
Write $\frac{7 + 3 2}{3 + 2 2}$ in the form $a + b 2$ , where $a$ and $b$ are integers.
[3 marks]
Answer

Example 4

Solve $x 12 + 5 = x 3 + 17$ . Give your answer in the form $a 3$ .

Hint

12 = 23

. Collect the

x

terms on one side, then rationalise.

B4
Solve $x 27 - 4 = x 12 + 11$ . Give your answer in the form $a 3$ .
[3 marks]
Answer $x = \dots\dots$

Example 5

A rectangle is $(3 + 2)$ cm wide. Its area is $(8 + 52)$ cm². Work out its length. Give your answer in the form $a + b 2$ .

Not drawn accurately

Hint Length

=

area

\div

width. Write it as a fraction and rationalise with

3 - 2

B5
A triangle has a base of $(4 + 3)$ cm. Its area is $(7 + 53)$ cm². Work out its perpendicular height $h$ . Give your answer in the form $a + b 3$ .
[3 marks]
Not drawn accurately
Answer $h = \dots\dots$ cm

Practice

Level 2

C1
Simplify
(a)
$180$
[1 mark]
Answer
(b)
$6 \times 15$
[2 marks]
Answer
(c)
$\frac{84}{7}$
[1 mark]
Answer
Total for C1: 4 marks
C2
Rationalise the denominator and simplify.
(a)
$\frac{10}{5}$
[1 mark]
Answer
(b)
$\frac{9}{2 3}$
[2 marks]
Answer
Total for C2: 3 marks

C3
Write $\frac{1}{3 - 5} + \frac{1}{3 + 5}$ as a single fraction in its simplest form.
[3 marks]
Answer
C4
Use the expansion of $(a + b)^{3}$ to write $(3 - 1)^{3}$ in the form $p + q 3$ .
[3 marks]
Answer
C5
Expand and simplify $(6 + 2)^{2}$ . Give your answer in the form $a + b 3$ .
[2 marks]
Answer
C6
Show that $(25 - 3) (25 + 3)$ is an integer.
[2 marks]

Exam-style questions

AQA exam style

D1
Kai says, " $12 + 27 = 39$ ."
Explain why Kai is wrong. Write $12 + 27$ in its simplest form.
[3 marks]
Answer $\dots\dots$

D2
Rationalise and simplify $\frac{2 3 + 1}{3 - 1}$ . Give your answer in the form $\frac{a + b 3}{2}$ , where $a$ and $b$ are integers.
[4 marks]
Answer
D3
The diagram shows a right-angled triangle. The two shorter sides are $(3 + 5)$ cm and $(3 - 5)$ cm.
Not drawn accurately
(a)
Show that the longest side is $27$ cm.
[2 marks]
(b)
Work out the area of the triangle.
[2 marks]
Answer $\dots\dots$ cm²
(c)
Hence write down the perimeter of the triangle.
[1 mark]
Answer $\dots\dots$ cm
Total for D3: 5 marks
D4
Solve $2 x + 3 = x 3 + 5$ . Give your answer in the form $a + b 3$ , where $a$ and $b$ are integers.
[4 marks]
Answer $x = \dots\dots$
D5
Which of these is equal to $\frac{98}{2}$ ? Circle your answer. $7249796$
[1 mark]
D6
Show that $(32 - 2)^{2} + (32 + 2)^{2}$ is an integer. You must show your working.
[2 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Show that $\frac{1}{1 + 2} + \frac{1}{2 + 3} + \frac{1}{3 + 4} + \dots + \frac{1}{15 + 16} = 3$
[5 marks]
Hint 1 · What to try
Rationalise one of the fractions, say $\frac{1}{2 + 3}$ , and look at what you get.
Hint 2 · The first line
Multiply the top and bottom of $\frac{1}{n + n + 1}$ by $n + 1 - n$ . The bottom becomes $(n + 1) - n = 1$ , so each fraction is $n + 1 - n$ .
Hint 3 · The full method
The sum is $(2 - 1) + (3 - 2) + \dots + (16 - 15)$ . Every root except $16$ and $1$ cancels, leaving $4 - 1 = 3$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify, add and multiply surds.
I can rationalise a denominator with one or two terms.
I can solve an equation with surds and give an exact answer.
I can use surds in area and Pythagoras problems.

Answers and mark scheme

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