Questions: Rationalizing the denominator

Author

Maximilian Volmar

Summary
A selection of questions for the study guide on rationalizing the denominator.

Before attempting these questions, it is highly recommended that you read Guide: Rationalizing the denominator.

Q1

Rationalize the denominator for each of the following expressions. Provide your answers in their simplest form and with a positive denominator.

1.1. \(\quad \dfrac{5}{\sqrt{3}}\)

1.2. \(\quad \dfrac{7}{2\sqrt{5}}\)

1.3. \(\quad \dfrac{11}{4\sqrt{7}}\)

1.4. \(\quad \dfrac{8}{5\sqrt{6}}\)

1.5. \(\quad \dfrac{3\sqrt{2}}{\sqrt{5}}\)

1.6. \(\quad \dfrac{9}{\sqrt{10}}\)

1.7. \(\quad \dfrac{\sqrt{7}}{\sqrt{3}}\)

1.8. \(\quad \dfrac{\sqrt{2}}{\sqrt{6}}\)

1.9. \(\quad \dfrac{12}{\sqrt{11}}\)

1.10. \(\quad \dfrac{\sqrt{8}}{\sqrt{2}}\)

1.11. \(\quad \dfrac{15}{3\sqrt{7}}\)

1.12. \(\quad \dfrac{6\sqrt{3}}{\sqrt{10}}\)

1.13. \(\quad \dfrac{\sqrt{18}}{\sqrt{9}}\)

1.14. \(\quad \dfrac{2\sqrt{5}}{\sqrt{12}}\)

1.15. \(\quad \dfrac{4}{\sqrt{2}}\)

1.16. \(\quad \dfrac{10}{5\sqrt{13}}\)


Q2

Rationalize the denominator for each of the following expressions. Provide your answers in their simplest form and with a positive denominator.

2.1. \(\quad \dfrac{5}{2 + \sqrt{3}}\)

2.2. \(\quad \dfrac{7}{4 - \sqrt{2}}\)

2.3. \(\quad \dfrac{3}{\sqrt{5} + 1}\)

2.4. \(\quad \dfrac{\sqrt{7}}{\sqrt{3} - 1}\)

2.5. \(\quad \dfrac{2 + \sqrt{5}}{1 - \sqrt{2}}\)

2.6. \(\quad \dfrac{3\sqrt{2} + 5}{4 + \sqrt{6}}\)

2.7. \(\quad \dfrac{8}{3 - \sqrt{7}}\)

2.8. \(\quad \dfrac{6}{2 + \sqrt{5}}\)

2.9. \(\quad \dfrac{\sqrt{10}}{\sqrt{2} + 3}\)

2.10. \(\quad \dfrac{2\sqrt{3} + 5}{\sqrt{7} - 1}\)

2.11. \(\quad \dfrac{\sqrt{6} - \sqrt{2}}{2 + \sqrt{5}}\)

2.12. \(\quad \dfrac{4 + \sqrt{3}}{5 - \sqrt{7}}\)

2.13. \(\quad \dfrac{2}{4 - \sqrt{11}}\)

2.14. \(\quad \dfrac{\sqrt{8} + \sqrt{3}}{\sqrt{7} - 2}\)


Q3

3.1. \(\quad\) The denominator of the expression \(\dfrac{\sqrt{11}}{2\sqrt{3}+\sqrt{5}}\) is not of the form \(b + c\sqrt{d}\), where \(b\) and \(c\) are integers and \(d\) is an integer that is not a perfect square but you can still rationalize the denominator.

Prove that

\[ \dfrac{\sqrt{11}}{2\sqrt{3}+\sqrt{5}} = \dfrac{2\sqrt{33}-\sqrt{55}}{7} \]

3.2. \(\quad\) Rationalize the denominator of this expression: \(\dfrac{5-\sqrt{2}}{\sqrt{10}-\sqrt{3}}\)

Provide your answer in its simplest form and with a positive denominator.


After attempting the questions above, please click this link to find the answers.


Version history and licensing

v1.0: initial version created 01/25 by Maximilian Volmar.

This work is licensed under CC BY-NC-SA 4.0.

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