Question 1
Rationalise the denominator of the following expressions
Give all answers in their simplest form.
1(a) \dfrac{1}{\sqrt{5}}
ANSWER: Multiple Choice (Type 1)
A: 2\sqrt{5}
B: 5\sqrt{5}
C: \dfrac{{\sqrt{5}}}{5}
D: \dfrac{{\sqrt{5}}}{25}
Answer: C
Workings:
\dfrac{1}{\sqrt{5}} \times \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{{\sqrt{5}}}{5}
Marks = 1
1(b) \dfrac{\sqrt{7}}{\sqrt{3}}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{7}{3}
B: \dfrac{3\sqrt{7}}{3}
C: \dfrac{\sqrt{7}}{21}
D: \dfrac{{\sqrt{21}}}{3}
Answer: D
Workings:
\dfrac{\sqrt{7}}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{\sqrt{21}}{3}
Marks = 1
1(c) \dfrac{\sqrt{3} +1 }{\sqrt{6}}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{\sqrt{18} +\sqrt{6}}{6}
B: \dfrac{\sqrt{24}}{6}
C: \dfrac{3 + \sqrt{6}}{6}
D: \dfrac{1 + \sqrt{6}}{6}
Answer: A
Workings:
\dfrac{\sqrt{3} +1 }{\sqrt{6}} = \dfrac{(\sqrt{3} +1) }{\sqrt{6}}\times \dfrac{\sqrt{6}}{\sqrt{6}} = \dfrac{\sqrt{3 \times 6} +\sqrt{6} }{6} = \dfrac{\sqrt{18} +\sqrt{6}}{6}
Marks = 1
1(d) \dfrac{(\sqrt{18}+8) }{\sqrt{3}} =
ANSWER: Multiple Choice (Type 1)
A: \dfrac{3\sqrt{18} + 8\sqrt{3}}{3}
B: \dfrac{\sqrt78}{3}
C: \dfrac{3\sqrt{6} + 8\sqrt{3}}{3}
D: \dfrac{3\sqrt{18} + 3\sqrt{8}}{3}
Answer: C
Workings:
\dfrac{\sqrt{18} + 8}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{3\sqrt{6} + 8\sqrt{3}}{3}
Marks = 1
Question 2
Rationalise the denominator of the following expressions
2(a) \dfrac{1}{3-\sqrt{2}}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{3 + \sqrt{2}}{11}
B: \dfrac{3 + \sqrt{2}}{7}
C: \dfrac{3 - \sqrt{2}}{7}
D: \dfrac{3\sqrt{2} + 2}{7}
Answer: B
Workings:
\dfrac{1}{3-\sqrt{6}} =\dfrac{1}{3-\sqrt{2}} \times \dfrac{3+\sqrt{2}}{3+\sqrt{2}} = \dfrac{3 + \sqrt{2}}{3^2 - (\sqrt{2})^2} = \dfrac{3 + \sqrt{2}}{7}
Marks = 2
2(b) \dfrac{3}{\sqrt{6} + 3}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{3 - \sqrt{6}}{5}
B: 9 + 3\sqrt{6}
C: \dfrac{9 - 3\sqrt{6}}{15}
D: 3 - \sqrt{6}
Answer: D
Workings:
\dfrac{3}{\sqrt{6} + 3} \times \dfrac{3-\sqrt{6}}{3-\sqrt{6}} = \dfrac{9-3\sqrt{6}}{9 - 6} = \dfrac{9-3\sqrt{6}}{3} = 3 - \sqrt{6}
Marks = 2
2(c) \dfrac{10}{\sqrt{7} - 6}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{-60 -10\sqrt{7}}{29}
B: \dfrac{60 + 10\sqrt{7}}{43}
C: \dfrac{60 -10\sqrt{7}}{29}
D:\dfrac{-60 - 10\sqrt{7}}{43}
Answer: A
Workings:
\dfrac{10}{\sqrt{7} - 6} \times \dfrac{-\sqrt{7}-6}{-\sqrt{7}-6} = \dfrac{-60 - 10\sqrt{7}}{-7 + 36} = \dfrac{-60 - 10\sqrt{7}}{29}
Marks = 2
Question 3
Rationalise the denominator of the following expressions
3(a) \dfrac{\sqrt{3} + 1}{\sqrt{5} + 2}
ANSWER: Multiple Choice (Type 1)
A: 2 + 2\sqrt{3} - \sqrt{15} - \sqrt{5}
B: \dfrac{2 + 2\sqrt{3} - \sqrt{15} - \sqrt{5}}{9}
C: \dfrac{\sqrt{15} + \sqrt{5} -2\sqrt{3} - 2}{9}
D: \sqrt{15} + \sqrt{5} -2\sqrt{3} - 2
Answer:D
Workings:
\dfrac{\sqrt{3} + 1}{\sqrt{5} + 2} \times \dfrac{-\sqrt{5} + 2}{-\sqrt{5} + 2} = \dfrac{-\sqrt{15} -\sqrt{5} + 2\sqrt{3} + 2}{-5 + 4}
= \sqrt{15} + \sqrt{5} -2\sqrt{3} - 2
Marks = 2
3(b) \dfrac{3 + \sqrt{2}}{\sqrt{6} + 3}
ANSWER: Multiple Choice (Type 1)
A:\dfrac{-3\sqrt{6} + 9 + 3\sqrt{2} - 2\sqrt{3}}{15}
B: = -\sqrt{6} + 3 + \sqrt{2} - \dfrac{2}{3}\sqrt{3}
C: \dfrac{3\sqrt{6} + 2\sqrt{3} - 9 - 3\sqrt{2}}{15}
D: = \sqrt{6} + \dfrac{2}{3}\sqrt{3} - 3 - \sqrt{2}
Answer: B
Workings:
\dfrac{3 + \sqrt{2}}{\sqrt{6} + 3} \times \dfrac{-\sqrt{6} + 3}{-\sqrt{6} + 3}= \dfrac{-3\sqrt{6} + 9 + 3\sqrt{2} -2\sqrt{3}}{-6 + 9}
= -\sqrt{6} + 3 + \sqrt{2} - \dfrac{2}{3}\sqrt{3}
Marks = 2
3(c) \dfrac{\sqrt{21} + 7}{\sqrt{21} - 7}
ANSWER: Multiple Choice (Type 1)
A: - \dfrac{5 + \sqrt{21}}{5}
B: \dfrac{5 + \sqrt{21}}{5}
C: -\dfrac{5 + \sqrt{21}}{2}
D: \dfrac{5 - \sqrt{21}}{2}
Answer: C
Workings:
\dfrac{\sqrt{21} + 7}{\sqrt{21} - 7} \times \dfrac{\sqrt{21} + 7}{\sqrt{21} + 7^2}
= \dfrac{(\sqrt{21}^2 + 2 \times 7 \times \sqrt{21} + 7^2}{(\sqrt{21})^2 - 7^2} = \dfrac{70 + 14\sqrt{21}}{-28}
= -\dfrac{5 + \sqrt{21}}{2}
Marks = 3
Question 4
Rationalise the denominators of the following expressions and simplify where possible
4(a) \dfrac{8 + \sqrt{5}}{4 - \sqrt{5}}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{37 - 13\sqrt{5}}{11}
B: \dfrac{37 + 13\sqrt{5}}{11}
C: \dfrac{37 + 13\sqrt{5}}{21}
D: -\dfrac{37 + 13\sqrt{5}}{21}
Answer: B
Workings:
\dfrac{8 + \sqrt{5}}{4 - \sqrt{5}} \times \dfrac{4 + \sqrt{5}}{4 + \sqrt{5}} = \dfrac{32 + 8\sqrt{5} + 4\sqrt{5} + 5}{16-5} = \dfrac{37 + 13\sqrt{5}}{11}Marks = 2
4(b) \dfrac{1}{(5-\sqrt{2})^2}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{27 + 10\sqrt{2}}{529}
B: -\dfrac{27 + 10\sqrt{2}}{529}
C: =\dfrac{27 + 10\sqrt{2}}{729}
D: =-\dfrac{27 + 10\sqrt{2}}{729}
Answer: A
Workings:
\dfrac{1}{(5-\sqrt{2})^2} \times \dfrac{(5+\sqrt{2})^2}{(5+\sqrt{2})^2}
= \dfrac{5^2 +\space2 \times 5 \times \sqrt{2} + (\sqrt{2})^2}{(5^2 - (\sqrt{2})^2)^2} = \dfrac{27 + 10\sqrt{2}}{(25-2)^2}
=\dfrac{27 + 10\sqrt{2}}{529}
Marks = 3
Question 5
Simplify,
\dfrac{6-5\sqrt{5}}{3\sqrt{5} - 2}
Give your answer in the form a + b\sqrt{5}, where a and b are rational numbers.
Answer: Multiple Answers (Type 1)
Answer: a = -\dfrac{63}{41}, b = \dfrac{8}{41}
Workings:
\dfrac{6-5\sqrt{5}}{3\sqrt{5} - 2} \times \dfrac{3\sqrt{5} + 2}{3\sqrt{5} + 2}
= \dfrac{18\sqrt{5} + 12 - 10\sqrt{5} - 15 \times 5}{(3\sqrt{5})^2 - 2^2} = \dfrac{-63 + 8\sqrt{5}}{41}
= -\dfrac{63}{41} + \dfrac{8}{41}\sqrt{5}
Marks = 4
Question 6
The following expression can be written in the form k\sqrt{a}, where k and a are integers.
\dfrac{4}{3}\sqrt{\dfrac{300}{4}} + \dfrac{10}{\sqrt{3}}
Work out the value of k and a.
ANSWER: Multiple Answers (Type 1)
Answer: k = 10, a = 3
Workings:
\dfrac{4}{3}\sqrt{\dfrac{300}{4}} + \dfrac{10}{\sqrt{3}} = \dfrac{4}{3}\sqrt{75} + \dfrac{10}{\sqrt{3}}
= \dfrac{4}{3}\sqrt{25 \times 3} + \dfrac{10}{\sqrt{3}}
= \dfrac{4 \times 5}{3}\sqrt{3} + \dfrac{10}{\sqrt{3}} = \dfrac{20}{3}\sqrt{3} + \dfrac{10}{\sqrt{3}} = \dfrac{20\sqrt{3}}{3} + \dfrac{10\sqrt{3}}{3}
= 10\sqrt{3}
Marks = 4
Question 7
The following expression can be written as \dfrac{a}{b}\sqrt{c}, where a, b and c are all integers:
\bigg(\dfrac{4}{3}\bigg)^{\dfrac{1}{2}} +\bigg (\dfrac{1}{3}\bigg)^{-\dfrac{1}{2}}
Work out the values of a, b and c.
ANSWER: Multiple Answers (Type 1)
Answer: a = 5, b = 3, c = 3
Workings:
\bigg(\dfrac{4}{3}\bigg)^{\dfrac{1}{2}} + \bigg(\dfrac{1}{3}\bigg)^{-\dfrac{1}{2}} = \dfrac{\sqrt{4}}{\sqrt{3}} + \dfrac{\sqrt{3}}{\sqrt{1}} = \dfrac{\sqrt{4}}{\sqrt{3}} + \bigg(\dfrac{\sqrt{3}}{\sqrt{1}} \times \dfrac{\sqrt{3}}{\sqrt{3}}\bigg)
\dfrac{\sqrt{4}}{\sqrt{3}} + \dfrac{3}{\sqrt{3}} = \dfrac{2}{\sqrt{3}} + \dfrac{3}{\sqrt{3}} = \dfrac{5}{\sqrt{3}}
= \dfrac{5}{3}\sqrt{3}
Marks = 3
Question 8
The following expression can be written in the form \dfrac{a\sqrt{3}}{b} where a and b are integers:
\sqrt{4\dfrac{12}{9}} + \bigg(\dfrac{1}{3}\bigg)^{\dfrac{1}{2}}
Work out the values of a and b.
ANSWER: Multiple Answers (Type 1)
Answer: a = 5, 3
Workings:
\sqrt{4\dfrac{12}{9}} = \sqrt{\dfrac{48}{9}}
\bigg(\dfrac{1}{3}\bigg)^{\dfrac{1}{2}} = \sqrt{\dfrac{1}{3}}
\sqrt{\dfrac{48}{9}} + \sqrt{\dfrac{1}{3}} = \dfrac{\sqrt{48}}{\sqrt{9}} + \dfrac{1}{\sqrt{3}} = \dfrac{4\sqrt{3}}{3} + \dfrac{1}{\sqrt{3}}
\dfrac{4\sqrt{3}}{3} + \dfrac{1}{\sqrt{3}} = \dfrac{4\sqrt{3}}{3} + \dfrac{\sqrt{3}}{3} = \dfrac{5\sqrt{3}}{3}
Marks = 4