RATIONALIZING DENOMINATORS

RATIONALIZING DENOMINATORS — Practice Questions

14 multiple-choice questions on rationalizing denominators.
Try to solve each question independently before checking the answer key.

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Instructions:
Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.

Quick Notes — Rationalizing Denominators:
For \(\dfrac{a}{\sqrt{b}}\): multiply by \(\dfrac{\sqrt{b}}{\sqrt{b}}\) → \(\dfrac{a\sqrt{b}}{b}\)

For \(\dfrac{a}{b+\sqrt{c}}\): multiply by \(\dfrac{b-\sqrt{c}}{b-\sqrt{c}}\) → \(\dfrac{a(b-\sqrt{c})}{b^2-c}\)

For \(\dfrac{a}{\sqrt{b}+\sqrt{c}}\): multiply by \(\dfrac{\sqrt{b}-\sqrt{c}}{\sqrt{b}-\sqrt{c}}\) → \(\dfrac{a(\sqrt{b}-\sqrt{c})}{b-c}\)

Problem 1

Simplified form of \(\dfrac{\sqrt{3}}{\sqrt{6}}\) is...

A. \(\dfrac{\sqrt{3}}{2}\) B. \(\dfrac{\sqrt{2}}{2}\) C. \(\dfrac{\sqrt{6}}{3}\) D. \(\dfrac{\sqrt{3}}{3}\) E. \(\dfrac{\sqrt{6}}{2}\)

Problem 2

Simplified form of \(\dfrac{4}{\sqrt{12}}\) is...

A. \(\dfrac{2}{3}\sqrt{5}\) B. \(\dfrac{2}{3}\sqrt{3}\) C. \(\dfrac{2}{5}\sqrt{5}\) D. \(\dfrac{3}{2}\sqrt{2}\) E. \(\dfrac{3}{2}\sqrt{3}\)

Problem 3

Simplified form of \(\dfrac{\sqrt{12}}{4\sqrt{3}}\) is...

A. \(\dfrac{1}{2}\sqrt{3}\) B. \(\dfrac{1}{2}\sqrt{2}\) C. \(\dfrac{1}{2}\) D. \(\dfrac{1}{3}\sqrt{2}\) E. \(\dfrac{1}{3}\)

Problem 4

Simplified form of \(\dfrac{10}{\sqrt{6}+\sqrt{2}}\) is...

A. \(\dfrac{2\sqrt{6}-3\sqrt{2}}{2}\) B. \(\dfrac{3\sqrt{6}-2\sqrt{2}}{2}\) C. \(\dfrac{5\sqrt{6}-5\sqrt{2}}{2}\) D. \(\dfrac{\sqrt{6}-\sqrt{2}}{2}\) E. \(\dfrac{\sqrt{6}-\sqrt{2}}{3}\)

Problem 5

Simplified form of \(\dfrac{\sqrt{3}}{\sqrt{8}-\sqrt{6}}\) is...

A. \(\dfrac{2\sqrt{6}+3\sqrt{2}}{2}\) B. \(\dfrac{3\sqrt{6}-4\sqrt{2}}{2}\) C. \(2\sqrt{6}+2\sqrt{2}\) D. \(\dfrac{4\sqrt{6}+3\sqrt{2}}{2}\) E. \(\dfrac{2\sqrt{2}+\sqrt{6}}{4}\)

Problem 6

Simplified form of \(\dfrac{\sqrt{5}}{\sqrt{15}-\sqrt{10}}\) is...

A. \(\dfrac{1}{2}(\sqrt{3}+\sqrt{2})\) B. \(\dfrac{1}{3}(\sqrt{3}+\sqrt{2})\) C. \(\sqrt{3}+\sqrt{2}\) D. \(\dfrac{1}{2}(\sqrt{5}+\sqrt{3})\) E. \(\dfrac{1}{2}(\sqrt{5}+\sqrt{3})\)

Problem 7

Simplified form of \(\dfrac{\sqrt{3}}{4-\sqrt{12}}\) is...

A. \(\dfrac{\sqrt{3}+3}{2}\) B. \(\dfrac{\sqrt{3}+2}{2}\) C. \(\dfrac{2\sqrt{3}+2}{2}\) D. \(\dfrac{3\sqrt{3}+3}{2}\) E. \(\dfrac{2\sqrt{3}+3}{2}\)

Problem 8

Simplified form of \(\dfrac{2\sqrt{6}}{3\sqrt{2}-4}\) is...

A. \(3\sqrt{3}+4\sqrt{6}\) B. \(6\sqrt{3}+8\sqrt{6}\) C. \(3\sqrt{3}+8\sqrt{6}\) D. \(6\sqrt{3}+4\sqrt{6}\) E. \(3\sqrt{3}+\sqrt{6}\)

Problem 9

Simplified form of \(\dfrac{\sqrt{6}-\sqrt{3}}{\sqrt{2}+\sqrt{8}}\) is...

A. \(\dfrac{2\sqrt{3}-5\sqrt{6}}{6}\) B. \(\dfrac{\sqrt{3}-3\sqrt{6}}{6}\) C. \(\dfrac{8\sqrt{3}+\sqrt{6}}{6}\) D. \(\dfrac{2\sqrt{3}-\sqrt{6}}{6}\) E. \(\dfrac{\sqrt{3}-2\sqrt{6}}{3}\)

Problem 10

Simplified form of \(\dfrac{\sqrt{5}+2}{\sqrt{5}-2}\) is...

A. \(\dfrac{9+2\sqrt{5}}{3}\) B. \(\dfrac{7+2\sqrt{5}}{3}\) C. \(9+2\sqrt{5}\) D. \(7+2\sqrt{5}\) E. \(9+4\sqrt{5}\)

Problem 11

Simplified form of \(\dfrac{12}{\sqrt{7}+\sqrt{3}}\) is...

A. \(3(\sqrt{7}+\sqrt{3})\) B. \(3(\sqrt{3}-\sqrt{7})\) C. \(3(\sqrt{7}-\sqrt{3})\) D. \(5(\sqrt{7}+\sqrt{3})\) E. \(5(\sqrt{7}-\sqrt{3})\)

Problem 12

Simplified form of \(\dfrac{3}{2+\sqrt{3}}\) is...

A. \(3(2+\sqrt{3})\) B. \(3(2-\sqrt{3})\) C. \(2(2+\sqrt{3})\) D. \(2(2-\sqrt{3})\) E. \(2-\sqrt{3}\)

Problem 13

Simplified form of \(\dfrac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}\) is...

A. \(6+2\sqrt{6}\) B. \(6-2\sqrt{6}\) C. \(5+2\sqrt{6}\) D. \(5-2\sqrt{6}\) E. \(2\sqrt{6}-5\)

Problem 14

If \(\dfrac{\frac{1}{2}-\frac{1}{\sqrt{5}}}{\frac{1}{2}+\frac{1}{\sqrt{5}}}=a+b\sqrt{5}\), then the value of \(a+b\) is...

A. \(5\) B. \(4\) C. \(3\) D. \(2\) E. \(1\)