Questions: 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.