A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 1: Algebraic expressions · Lesson 3

Surds and rationalising denominators

73 marks

Hints online, answers free with an account:
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Date

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Quick questions to start.

A1
Needed today
Expand and simplify $(2 x - 5) (2 x + 5)$ .
[2 marks]
Answer
A2
GCSE Higher
Find the largest square factor of $180$ .
[2 marks]
Answer
A3
Last lesson
Factorise $6 x^{2} + x - 15$ .
[2 marks]
Answer
A4
From chapter 1
Evaluate $(\frac{243}{32})^{- \frac{2}{5}}$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Simplify (a) $98$ (b) $300 - 75 + 212$ .

Hint Find the largest square factor:

98 = 49 \times 2

B1
Simplify (a) $245$ (b) $288 + 50 - 332$ .
[3 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Expand and simplify $(3 - 5) (4 + 25)$ .

Hint Four products, as with

(x + a) (x + b)

. Then

5 \times 5 = 5

B2
Expand and simplify (a) $6 (3 + 8)$ (b) $(5 + 23)^{2}$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

Rationalise the denominator and simplify (a) $\frac{14}{7}$ (b) $\frac{6 - 2}{3 + 2}$ .

Hint In (b) multiply top and bottom by

3 - 2

. The bottom becomes

9 - 2

B3
Rationalise the denominator and simplify (a) $\frac{18}{6}$ (b) $\frac{5 + 3}{4 - 3}$ , giving (b) in the form $p + q 3$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 4

Rationalise the denominator of $\frac{7 + 3}{7 - 3}$ .

Hint Multiply top and bottom by

7 + 3

. The bottom is

7 - 3

B4
Rationalise the denominator and simplify $\frac{11 - 5}{11 + 5}$ .
[3 marks]
Answer

Example 5

Rationalise the denominator of $\frac{1}{( 3 + 5 ) ^{2}}$ .

Hint Expand the bottom first:

(3 + 5)^{2} = 14 + 65

. Then multiply top and bottom by

14 - 65

B5
Rationalise the denominator and simplify $\frac{6}{( 5 - 1 ) ^{2}}$ .
[3 marks]
Answer

Practice

A-Level

C1
A square has sides of length $(2 + 5)$ cm. A rectangle has the same area as the square, and one of its sides is $5$ cm. Find the length of the other side of the rectangle, in the form $a + b 5$ where $a$ and $b$ are rational.
[3 marks]
Not drawn accurately
Answer $\dots\dots$ cm

C2
Write $\frac{4 - 2}{3 2 - 2}$ in the form $p + q 2$ , where $p$ and $q$ are rational numbers.
[4 marks]
Answer
C3
Solve $5 x - 3 = x 3 + 6$ . Give your answer in the form $a + b 3$ , where $a$ and $b$ are rational.
[4 marks]
Answer
C4
$p = 5 + 26$ and $q = 5 - 26$ . Show that $\frac{1}{p} = q$ , and find the value of $p^{2} + q^{2}$ .
[4 marks]
Answer $p^{2} + q^{2} = \dots\dots$

Exam-style questions

Edexcel exam style

D1
Ana is asked to write $\frac{10}{4 - 6}$ in the form $a + b 6$ . She writes: $\frac{10}{4 - 6} \times \frac{4 + 6}{4 + 6} = \frac{40 + 10 6}{16 + 6} = \frac{20 + 5 6}{11}$ .
Ana has made a mistake.
(a)
Explain the mistake Ana has made.
[1 mark]
(b)
Write $\frac{10}{4 - 6}$ correctly in the form $a + b 6$ .
[2 marks]
Answer
(c)
Hence find the exact value of $\frac{10}{4 - 6} - \frac{10}{4 + 6}$ .
[2 marks]
Answer
Total for D1: 5 marks

D2
Do not use a calculator in this question.
(a)
Show that $(6 + 2)^{2} = 8 + 43$ .
[2 marks]
(b)
Hence write $8 + 43$ in the form $a + b$ , where $a$ and $b$ are integers.
[1 mark]
Answer
(c)
Hence write $\frac{4}{8 + 4 3}$ in the form $c - d$ , where $c$ and $d$ are integers.
[3 marks]
Answer
Total for D2: 6 marks
D3
The diagram shows a rectangle $A B C D$ . $A B = (3 + 5)$ cm and the area of the rectangle is $(14 + 65)$ cm².
Not drawn accurately
(a)
Show that $B C = (3 + 5)$ cm.
[3 marks]
(b)
Hence find the perimeter of $A B C D$ , in the form $a + b 5$ .
[1 mark]
Answer $\dots\dots$ cm
(c)
Show that $A C^{2} = 28 + 125$ .
[2 marks]
(d)
A circle passes through $A$ , $B$ , $C$ and $D$ . Find the exact area of the circle, in the form $π (p + q 5)$ .
[2 marks]
Answer $\dots\dots$ cm²
Total for D3: 8 marks
D4
In triangle $P QR$ , angle $P QR = 9 0^{\circ}$ , $P Q = (23 + 1)$ cm and $QR = (23 - 1)$ cm.
Not drawn accurately
(a)
Show that $P R = 26$ cm.
[3 marks]
(b)
Find the exact area of triangle $P QR$ .
[2 marks]
Answer $\dots\dots$ cm²
(c)
The perpendicular from $Q$ to $P R$ has length $h$ cm. Hence find $h$ , giving your answer in the form $\frac{a 26}{b}$ where $a$ and $b$ are integers.
[3 marks]
Answer $h = \dots\dots$
Total for D4: 8 marks

Extension

Stretch

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

E1
The number $5 - 2$ is a root of an equation of the form $x^{4} + p x^{2} + q = 0$ , where $p$ and $q$ are integers. Find $p$ and $q$ .
[6 marks]
Hint 1 · What to try
Let $x = 5 - 2$ and square it. One surd is left.
Hint 2 · The first line
$x^{2} = 7 - 210$ . Get the surd on its own: $x^{2} - 7 = - 210$ . Square again.
Hint 3 · The full method
$(x^{2} - 7)^{2} = 40$ , so $x^{4} - 14 x^{2} + 49 = 40$ and $x^{4} - 14 x^{2} + 9 = 0$ .
Answer $p = \dots\dots$ $q = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify a surd and collect like surds.
I can expand brackets that have surds in them.
I can rationalise a denominator with one or two surds.
I can give exact answers for lengths, areas and equations.

Answers and mark scheme

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