Question 1
LEVEL 6
Rationalise the denominator of \dfrac{7}{3+\sqrt{2}} and simplify your answer.
Select the correct answer from the list below:
A: \dfrac{21+7\sqrt{2}}{11+6\sqrt{2}}
B: 3 - \sqrt{2}
C: \dfrac{21-7\sqrt{2}}{5}
D: \dfrac{21-7\sqrt{2}}{11}
CORRECT ANSWER: B: 3 - \sqrt{2}
WORKED SOLUTION:
\dfrac{7}{3+\sqrt{2}}\dfrac{7}{3+\sqrt{2}} \times \dfrac{(3 - \sqrt{2})}{(3 - \sqrt{2})}
\dfrac{7 \times 3 - 7\sqrt{2}}{(3 + \sqrt{2})(3 - \sqrt{2})}
\dfrac{21 - 7\sqrt{2}}{9 - 3\sqrt{2} + 3\sqrt{2} - \sqrt{2} \times \sqrt{2}}
\dfrac{21 - 7\sqrt{2}}{9 - 2}
Divide the top and bottom by 7 to simplify:
3 - \sqrt{2}Question 2
LEVEL 6
Rationalise the denominator of \dfrac{2}{\sqrt{3}}
Select the correct answer from the list below:
A: \dfrac{\sqrt{3}}{2}
B: \dfrac{\sqrt{3}}{3}
C: \dfrac{2\sqrt{3}}{3}
D: \dfrac{2}{3}
CORRECT ANSWER: C: \dfrac{2\sqrt{3}}{3}
WORKED SOLUTION:
Multiply the numerator and denominator by the same surd to rationalise the denominator
\dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}
\dfrac{2\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}} \dfrac{2\sqrt{3}}{3}Question 3
LEVEL 6
Rationalise the denominator of \dfrac{1}{\sqrt{7}}
Select the correct answer from the list below:
A: \dfrac{7}{7}
B: \dfrac{7}{\sqrt{7}}
C: \dfrac{\sqrt{7}}{\sqrt{7}}
D: \dfrac{\sqrt{7}}{7}
CORRECT ANSWER: D: \dfrac{\sqrt{7}}{7}
WORKED SOLUTION:
\dfrac{1}{\sqrt{7}} \times \dfrac{\sqrt{7}}{\sqrt{7}}
\dfrac{1\times\sqrt{7}}{\sqrt{7}\times\sqrt{7}}
\dfrac{\sqrt{7}}{7}
Question 4
LEVEL 6
Rationalise the denominator of \dfrac{21}{3\sqrt{5}}
Select the correct answer from the list below:
A: \dfrac{21\sqrt{5}}{15}
B: \dfrac{\sqrt{5}}{5}
C: \dfrac{5}{21\sqrt{5}}
D: \dfrac{21\sqrt{5}}{5}
CORRECT ANSWER: A:\dfrac{21\sqrt{5}}{15}
WORKED SOLUTION:
\dfrac{21}{3\sqrt{5}}\times \dfrac{\sqrt{5}}{\sqrt{5}}
\dfrac{21\times \sqrt{5}}{3\times \sqrt{5}\times \sqrt{5}}
\dfrac{21\sqrt{5}}{5\times3}
\dfrac{21\sqrt{5}}{15}
Question 5
LEVEL 6
Rationalise the denominator of \dfrac{2}{2+\sqrt{3}}
Select the correct answer from the list below:
A: 4-2\sqrt{3}
B: 2\sqrt{3}
C: \dfrac{4-2\sqrt{3}}{3}
D: \dfrac{2\sqrt{3}}{3}
CORRECT ANSWER: A: 4-2\sqrt{3}
WORKED SOLUTION:
\dfrac{2}{2+\sqrt{3}}
\dfrac{2}{2+\sqrt{3}}\times \dfrac{2-\sqrt{3}}{2-\sqrt{3}}
=\dfrac{2\times 2-2\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}
=\dfrac{4-2\sqrt{3}}{4-2\sqrt{3}+2\sqrt{3}-\sqrt{3}\times \sqrt{3}}
=\dfrac{4-2\sqrt{3}}{4-3}
=\dfrac{4-2\sqrt{3}}{1}
=4-2\sqrt{3}