Question 1

LEVEL 6

Rationalise the denominator of 73+2\dfrac{7}{3+\sqrt{2}} and simplify your answer.

Select the correct answer from the list below:

A: 21+7211+62\dfrac{21+7\sqrt{2}}{11+6\sqrt{2}}

B: 3−23 - \sqrt{2}

C: 21−725\dfrac{21-7\sqrt{2}}{5}

D: 21−7211\dfrac{21-7\sqrt{2}}{11}

CORRECT ANSWER: B: 3−23 - \sqrt{2}

WORKED SOLUTION:

73+2\dfrac{7}{3+\sqrt{2}}

 

73+2×(3−2)(3−2)\dfrac{7}{3+\sqrt{2}} \times \dfrac{(3 - \sqrt{2})}{(3 - \sqrt{2})}

 

7×3−72(3+2)(3−2)\dfrac{7 \times 3 - 7\sqrt{2}}{(3 + \sqrt{2})(3 - \sqrt{2})}

 

21−729 −32+32−2×2\dfrac{21 - 7\sqrt{2}}{9  - 3\sqrt{2} + 3\sqrt{2} - \sqrt{2} \times \sqrt{2}}

 

21−729−2\dfrac{21 - 7\sqrt{2}}{9 - 2}

 

Divide the top and bottom by 77 to simplify:

3−23 - \sqrt{2}

Question 2

LEVEL 6

Rationalise the denominator of 23\dfrac{2}{\sqrt{3}}

Select the correct answer from the list below:

A:32 \dfrac{\sqrt{3}}{2}

B: 33 \dfrac{\sqrt{3}}{3}

C: 233 \dfrac{2\sqrt{3}}{3}

D: 23 \dfrac{2}{3}

CORRECT ANSWER: C: 233 \dfrac{2\sqrt{3}}{3}

WORKED SOLUTION:

Multiply the numerator and denominator by the same surd to rationalise the denominator

23×33\dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}

2×33×3 \dfrac{2\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}} 233 \dfrac{2\sqrt{3}}{3}

Question 3

LEVEL 6

Rationalise the denominator of 17\dfrac{1}{\sqrt{7}}

Select the correct answer from the list below:

A: 77 \dfrac{7}{7}

B:77 \dfrac{7}{\sqrt{7}}

C: 77 \dfrac{\sqrt{7}}{\sqrt{7}}

D: 77 \dfrac{\sqrt{7}}{7}

CORRECT ANSWER:   D: 77 \dfrac{\sqrt{7}}{7}

WORKED SOLUTION:

17×77\dfrac{1}{\sqrt{7}} \times \dfrac{\sqrt{7}}{\sqrt{7}}

 

1×77×7 \dfrac{1\times\sqrt{7}}{\sqrt{7}\times\sqrt{7}}

 

77 \dfrac{\sqrt{7}}{7}

Question 4

LEVEL 6

Rationalise the denominator of 2135\dfrac{21}{3\sqrt{5}}

Select the correct answer from the list below:

A: 21515\dfrac{21\sqrt{5}}{15}

B: 55\dfrac{\sqrt{5}}{5}

C: 5215\dfrac{5}{21\sqrt{5}}

D: 2155\dfrac{21\sqrt{5}}{5}

CORRECT ANSWER:  A:21515\dfrac{21\sqrt{5}}{15}

WORKED SOLUTION:

2135×55\dfrac{21}{3\sqrt{5}}\times \dfrac{\sqrt{5}}{\sqrt{5}}

21×53×5×5\dfrac{21\times \sqrt{5}}{3\times \sqrt{5}\times \sqrt{5}}

2155×3\dfrac{21\sqrt{5}}{5\times3}

21515\dfrac{21\sqrt{5}}{15}

Question 5

LEVEL 6

Rationalise the denominator of 22+3\dfrac{2}{2+\sqrt{3}}

Select the correct answer from the list below:

A: 4−234-2\sqrt{3}

B: 23 2\sqrt{3}

C: 4−233\dfrac{4-2\sqrt{3}}{3}

D: 233\dfrac{2\sqrt{3}}{3}

CORRECT ANSWER:  A: 4−234-2\sqrt{3}

WORKED SOLUTION:

22+3\dfrac{2}{2+\sqrt{3}}

22+3×2−32−3\dfrac{2}{2+\sqrt{3}}\times \dfrac{2-\sqrt{3}}{2-\sqrt{3}}

=2×2−23(2+3)(2−3)=\dfrac{2\times 2-2\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}

=4−234−23+23−3×3=\dfrac{4-2\sqrt{3}}{4-2\sqrt{3}+2\sqrt{3}-\sqrt{3}\times \sqrt{3}}

=4−234−3=\dfrac{4-2\sqrt{3}}{4-3}

=4−231=\dfrac{4-2\sqrt{3}}{1}

=4−23=4-2\sqrt{3}