Rationalising Surds (D) Worksheet | Printable Maths Worksheets - Free Printable

Here are the solutions for the worksheet. I have worked through each problem step-by-step to ensure accuracy.

Section A: Rationalise the denominator

*Method: Multiply the top and bottom by the square root in the denominator.*

1) $\frac{3}{\sqrt{2}}$
Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$:
$\frac{3 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{3\sqrt{2}}{2}$

2) $\frac{11}{\sqrt{5}}$
Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$:
$\frac{11 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{11\sqrt{5}}{5}$

3) $\frac{9}{\sqrt{7}}$
Multiply by $\frac{\sqrt{7}}{\sqrt{7}}$:
$\frac{9 \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{9\sqrt{7}}{7}$

4) $\frac{4}{\sqrt{11}}$
Multiply by $\frac{\sqrt{11}}{\sqrt{11}}$:
$\frac{4 \times \sqrt{11}}{\sqrt{11} \times \sqrt{11}} = \frac{4\sqrt{11}}{11}$

5) $\frac{20}{\sqrt{13}}$
Multiply by $\frac{\sqrt{13}}{\sqrt{13}}$:
$\frac{20 \times \sqrt{13}}{\sqrt{13} \times \sqrt{13}} = \frac{20\sqrt{13}}{13}$

6) $\frac{15}{\sqrt{3}}$
Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$:
$\frac{15\sqrt{3}}{3}$. Simplify $15 \div 3 = 5$.
Answer: $5\sqrt{3}$

7) $\frac{14}{\sqrt{2}}$
Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$:
$\frac{14\sqrt{2}}{2}$. Simplify $14 \div 2 = 7$.
Answer: $7\sqrt{2}$

8) $\frac{20}{\sqrt{5}}$
Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$:
$\frac{20\sqrt{5}}{5}$. Simplify $20 \div 5 = 4$.
Answer: $4\sqrt{5}$

9) $\frac{22}{\sqrt{11}}$
Multiply by $\frac{\sqrt{11}}{\sqrt{11}}$:
$\frac{22\sqrt{11}}{11}$. Simplify $22 \div 11 = 2$.
Answer: $2\sqrt{11}$

10) $\frac{8}{\sqrt{6}}$
Multiply by $\frac{\sqrt{6}}{\sqrt{6}}$:
$\frac{8\sqrt{6}}{6}$. Simplify fraction $\frac{8}{6}$ to $\frac{4}{3}$.
Answer: $\frac{4\sqrt{6}}{3}$

11) $\frac{28}{3\sqrt{7}}$
Multiply by $\frac{\sqrt{7}}{\sqrt{7}}$:
$\frac{28\sqrt{7}}{3 \times 7} = \frac{28\sqrt{7}}{21}$.
Simplify fraction $\frac{28}{21}$ (divide both by 7) to $\frac{4}{3}$.
Answer: $\frac{4\sqrt{7}}{3}$

12) $\frac{18}{5\sqrt{3}}$
Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$:
$\frac{18\sqrt{3}}{5 \times 3} = \frac{18\sqrt{3}}{15}$.
Simplify fraction $\frac{18}{15}$ (divide both by 3) to $\frac{6}{5}$.
Answer: $\frac{6\sqrt{3}}{5}$

13) $\frac{2}{7\sqrt{4}}$
First, simplify $\sqrt{4} = 2$. The expression becomes $\frac{2}{7 \times 2} = \frac{2}{14}$.
Simplify $\frac{2}{14}$ to $\frac{1}{7}$.
Answer: $\frac{1}{7}$

14) $\frac{25}{3\sqrt{5}}$
Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$:
$\frac{25\sqrt{5}}{3 \times 5} = \frac{25\sqrt{5}}{15}$.
Simplify fraction $\frac{25}{15}$ (divide both by 5) to $\frac{5}{3}$.
Answer: $\frac{5\sqrt{5}}{3}$

15) $\frac{54}{9\sqrt{2}}$
First, simplify numbers: $54 \div 9 = 6$. Expression becomes $\frac{6}{\sqrt{2}}$.
Rationalise: $\frac{6\sqrt{2}}{2}$. Simplify $6 \div 2 = 3$.
Answer: $3\sqrt{2}$

16) $\frac{\sqrt{27}}{\sqrt{3}}$
Combine roots: $\sqrt{\frac{27}{3}} = \sqrt{9}$.
$\sqrt{9} = 3$.
Answer: $3$

17) $\frac{\sqrt{75}}{\sqrt{3}}$
Combine roots: $\sqrt{\frac{75}{3}} = \sqrt{25}$.
$\sqrt{25} = 5$.
Answer: $5$

18) $\frac{\sqrt{64}}{\sqrt{4}}$
Calculate roots: $\sqrt{64} = 8$ and $\sqrt{4} = 2$.
$\frac{8}{2} = 4$.
Answer: $4$

19) $\frac{\sqrt{98}}{\sqrt{2}}$
Combine roots: $\sqrt{\frac{98}{2}} = \sqrt{49}$.
$\sqrt{49} = 7$.
Answer: $7$

20) $\frac{\sqrt{24}}{\sqrt{6}}$
Combine roots: $\sqrt{\frac{24}{6}} = \sqrt{4}$.
$\sqrt{4} = 2$.
Answer: $2$

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Section B: Rationalise the denominators (Binomials)

*Method: Multiply top and bottom by the "conjugate" (change the sign in the middle).*

1) $\frac{11}{2 - \sqrt{3}}$
Multiply by conjugate $(2 + \sqrt{3})$:
Top: $11(2 + \sqrt{3}) = 22 + 11\sqrt{3}$
Bottom: $(2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$
Answer: $22 + 11\sqrt{3}$

2) $\frac{1}{2 - \sqrt{5}}$
Multiply by conjugate $(2 + \sqrt{5})$:
Top: $1(2 + \sqrt{5}) = 2 + \sqrt{5}$
Bottom: $(2 - \sqrt{5})(2 + \sqrt{5}) = 4 - 5 = -1$
Result: $\frac{2 + \sqrt{5}}{-1}$
Answer: $-2 - \sqrt{5}$

6) $\frac{12}{3 - \sqrt{3}}$
Multiply by conjugate $(3 + \sqrt{3})$:
Top: $12(3 + \sqrt{3}) = 36 + 12\sqrt{3}$
Bottom: $(3 - \sqrt{3})(3 + \sqrt{3}) = 9 - 3 = 6$
Simplify: $\frac{36 + 12\sqrt{3}}{6} = \frac{36}{6} + \frac{12\sqrt{3}}{6}$
Answer: $6 + 2\sqrt{3}$

7) $\frac{7}{1 + \sqrt{2}}$
Multiply by conjugate $(1 - \sqrt{2})$:
Top: $7(1 - \sqrt{2}) = 7 - 7\sqrt{2}$
Bottom: $(1 + \sqrt{2})(1 - \sqrt{2}) = 1 - 2 = -1$
Result: $\frac{7 - 7\sqrt{2}}{-1}$
Answer: $-7 + 7\sqrt{2}$ (or $7\sqrt{2} - 7$)

11) $\frac{4}{3 + \sqrt{11}}$
Multiply by conjugate $(3 - \sqrt{11})$:
Top: $4(3 - \sqrt{11}) = 12 - 4\sqrt{11}$
Bottom: $(3 + \sqrt{11})(3 - \sqrt{11}) = 9 - 11 = -2$
Simplify: $\frac{12 - 4\sqrt{11}}{-2} = \frac{12}{-2} - \frac{4\sqrt{11}}{-2}$
Answer: $-6 + 2\sqrt{11}$ (or $2\sqrt{11} - 6$)

12) $\frac{1}{\sqrt{3} - 1}$
Multiply by conjugate $(\sqrt{3} + 1)$:
Top: $1(\sqrt{3} + 1) = \sqrt{3} + 1$
Bottom: $(\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2$
Answer: $\frac{\sqrt{3} + 1}{2}$

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Final Answer:

Section A
1) $\frac{3\sqrt{2}}{2}$
2) $\frac{11\sqrt{5}}{5}$
3) $\frac{9\sqrt{7}}{7}$
4) $\frac{4\sqrt{11}}{11}$
5) $\frac{20\sqrt{13}}{13}$
6) $5\sqrt{3}$
7) $7\sqrt{2}$
8) $4\sqrt{5}$
9) $2\sqrt{11}$
10) $\frac{4\sqrt{6}}{3}$
11) $\frac{4\sqrt{7}}{3}$
12) $\frac{6\sqrt{3}}{5}$
13) $\frac{1}{7}$
14) $\frac{5\sqrt{5}}{3}$
15) $3\sqrt{2}$
16) $3$
17) $5$
18) $4$
19) $7$
20) $2$

Section B
1) $22 + 11\sqrt{3}$
2) $-2 - \sqrt{5}$
6) $6 + 2\sqrt{3}$
7) $-7 + 7\sqrt{2}$
11) $-6 + 2\sqrt{11}$
12) $\frac{\sqrt{3} + 1}{2}$