RADICAL EXPRESSIONS

RADICAL EXPRESSIONS — Practice Questions

44 multiple-choice questions on radical expressions.
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 — Radical Expressions:
\(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\)
\(\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}\)
\(\sqrt{(a+b)+2\sqrt{ab}} = \sqrt{a} + \sqrt{b}\)
\(\sqrt{(a+b)-2\sqrt{ab}} = \sqrt{a} - \sqrt{b}\) (for \(a \ge b\))
\((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b\)

Problem 1

If integers \(a\) and \(b\) satisfy \(\dfrac{\sqrt{5}-\sqrt{6}}{\sqrt{5}+\sqrt{6}}=a+b\sqrt{30}\), then \(ab=\cdots\)

A. \(-22\) B. \(-11\) C. \(-9\) D. \(2\) E. \(13\)

Problem 2

Simplified form of \(\dfrac{\sqrt{143}+\sqrt{165}+\sqrt{195}+13}{\sqrt{11}+2\sqrt{13}+\sqrt{15}}\) is...

A. \(\frac{1}{2}(\sqrt{15}+\sqrt{13})\) B. \(\frac{1}{2}\frac{\sqrt{15}-\sqrt{13}}{\sqrt{15}+\sqrt{13}}\) C. \(\frac{1}{2}(\sqrt{15}-\sqrt{11})\) D. \(\frac{1}{2}(\sqrt{15}+\sqrt{11})\) E. \(\frac{\sqrt{15}-\sqrt{11}}{\sqrt{15}+\sqrt{11}}\)

Problem 3

If \(\sqrt{0.3+\sqrt{0.08}}=\sqrt{a}+\sqrt{b}\), then \(\dfrac{1}{a}+\dfrac{1}{b}=\cdots\)

A. \(25\) B. \(20\) C. \(15\) D. \(10\) E. \(5\)

Problem 4

Rationalized form of \(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{1-\sqrt{2}}\) is...

A. \(-1-\frac{1}{2}\sqrt{2}\) B. \(-1-\sqrt{2}\) C. \(-\frac{1}{2}\sqrt{2}\) D. \(\frac{1}{2}\sqrt{2}\) E. \(2+\frac{1}{2}\sqrt{2}\)

Problem 5

Value of \(\dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\dfrac{1}{\sqrt{63}+\sqrt{64}}\) is...

A. \(10\) B. \(9\) C. \(8\) D. \(7\) E. \(6\)

Problem 6

\(\dfrac{\sqrt{18}-\sqrt{12}}{\sqrt{18}+\sqrt{12}}+\dfrac{5}{1+\sqrt{6}}=\cdots\)

A. \(\sqrt{6}\) B. \(1-\sqrt{6}\) C. \(\sqrt{2}+\sqrt{3}\) D. \(4-\sqrt{6}\) E. \(5-2\sqrt{6}\)

Problem 7

If \(r=\dfrac{20\sqrt{2}-25}{(10+20\sqrt{2})(2-\sqrt{2})}\), then \((4r-2)^2=\cdots\)

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

Problem 8

If \(\sqrt[4]{a}+\sqrt[4]{9}=\dfrac{1}{2-\sqrt{3}}\), then \(a=\cdots\)

A. \(2-\sqrt{2}\) B. \(2\) C. \(2+\sqrt{2}\) D. \(8\) E. \(16\)

Problem 9

Simplified form of \(\sqrt{\dfrac{\sqrt{41}+4}{\sqrt{41}-4}}-\sqrt{\dfrac{\sqrt{41}-4}{\sqrt{41}+4}}\) is...

A. \(-\frac{8}{5}\) B. \(0\) C. \(\frac{16}{5}\) D. \(\frac{8}{5}\) E. \(5\sqrt{41}\)

Problem 10

\(\sqrt{3-\sqrt{5}}+\sqrt{3+\sqrt{5}}=\cdots\)

A. \(2\sqrt{3}\) B. \(\sqrt{10}\) C. \(2\sqrt{2}\) D. \(\sqrt{11}\) E. \(3\sqrt{2}\)

Problem 11

\(\dfrac{5(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})^3}{2\sqrt{2}-\sqrt{3}}=\cdots\)

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

Problem 12

\(\sqrt{3+2\sqrt{2}}-\sqrt{2}=\cdots\)

A. \(4\sqrt{2}\) B. \(3+\sqrt{2}\) C. \(\sqrt{2}\) D. \(1\) E. \(0\)

Problem 13

\(\sqrt{\dfrac{8}{15}-2\sqrt{\dfrac{1}{15}}}=\cdots\)

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

Problem 14

The smallest positive integer \(n\) satisfying \(\sqrt{n}-\sqrt{n-1} < 0.01\) is...

A. \(2499\) B. \(2500\) C. \(2501\) D. \(10000\) E. no such integer

Problem 15

Simplified form of \(\sqrt{9+\sqrt{17}}-\sqrt{9-\sqrt{17}}\) is...

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

Problem 16

If \(\sqrt{7x^2-2x+432}+\sqrt{7x^2-2x-423}=285\), then \(\sqrt{7x^2-2x+432}-\sqrt{7x^2-2x-423}=\cdots\)

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

Problem 17

Value of \(\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\) is...

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

Problem 18

Simplified form of \(\dfrac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}\) is...

A. \(a+b-\sqrt{ab}\) B. \(a-b+\sqrt{ab}\) C. \(a+b+\sqrt{ab}\) D. \(a-b-\sqrt{ab}\) E. \(-a-b-\sqrt{ab}\)

Problem 19

Value of \(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}-3\) is...

A. \(-2\) B. \(-1\) C. \(1\) D. \(1.5\) E. \(2\)

Problem 20

Given \(a\) and \(b\) are positive integers with \(a > b\). If \(\sqrt{95+2\sqrt{2016}}=\sqrt{a}+\sqrt{b}\), then \(a-b=\cdots\)

A. \(25\) B. \(29\) C. \(31\) D. \(32\) E. \(35\)

Problem 21

\(a=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}\) and \(b=\sqrt{20+\sqrt{20+\sqrt{20+\cdots}}}\). Then \(a+b=\cdots\)

A. \(\sqrt{26}\) B. \(8\) C. \(2\sqrt{26}\) D. \(16\) E. \(26\)

Problem 22

Simplified form of \(\dfrac{2}{\sqrt{3}-\sqrt{2}}-\dfrac{1}{2-\sqrt{3}}-\dfrac{5}{\sqrt{8}-\sqrt{3}}\) is...

A. \(-10\) B. \(-5\) C. \(-2\) D. \(2\) E. \(10\)

Problem 23

\(\dfrac{\sqrt{45}+\sqrt{18}}{\sqrt{7+2\sqrt{10}}}=\cdots\)

A. \(3\) B. \(6\) C. \(9\) D. \(3\sqrt{2}\) E. \(2\sqrt{3}\)

Problem 24

Value of \(\sqrt{4+\sqrt{16+\sqrt{64+\sqrt{\cdots}}}}\) is...

A. \(2\) B. \(3\) C. \(4\) D. \(5\) E. \(6\)

Problem 25

Simplified form of \(78(\sqrt{17+12\sqrt{2}}+\sqrt{17-12\sqrt{2}})\) is...

A. \(234\) B. \(312\) C. \(468\) D. \(546\) E. \(624\)

Problem 26

Simplified form of \((\sqrt{52+6\sqrt{43}})^3-(\sqrt{52-6\sqrt{43}})^3\) is...

A. \(184\) B. \(288\) C. \(476\) D. \(828\) E. \(900\)

Problem 27

Simplified form of \(\sqrt{7+\sqrt{48}}\) is...

A. \(\sqrt{8}+\sqrt{7}\) B. \(\sqrt{7}+\sqrt{6}\) C. \(\sqrt{6}+1\) D. \(\sqrt{5}+\sqrt{2}\) E. \(\sqrt{4}+\sqrt{3}\)

Problem 28

\(\dfrac{5(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})^3}{2\sqrt{2}-\sqrt{3}}=\cdots\)

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

Problem 29

\(\dfrac{(\sqrt[6]{x^2})(\sqrt[3]{x^2\sqrt{x+1}})}{x\sqrt[6]{x+1}}=\cdots\)

A. \(x\sqrt{x+1}\) B. \(x\) C. \(1\) D. \(\dfrac{1}{\sqrt[8]{x^2}}\) E. \(\dfrac{x}{\sqrt{x+1}}\)

Problem 30

\(\dfrac{\sqrt{20}+\sqrt{8}}{\sqrt{2}+\sqrt{5}}=\cdots\)

A. \(2\) B. \(2\sqrt{10}\) C. \(3\) D. \(3\sqrt{10}\) E. \(4\)

Problem 31

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

A. \(5+2\sqrt{6}\) B. \(6+2\sqrt{6}\) C. \(5+3\sqrt{6}\) D. \(6+3\sqrt{6}\) E. \(7+2\sqrt{6}\)

Problem 32

\((\sqrt{2}-\sqrt{3})(\sqrt{3}-\sqrt{5})(\sqrt{5}-\sqrt{2})=\cdots\)

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

Problem 33

If \(\sqrt{9-6\sqrt{2}}=\sqrt{a}-\sqrt{b}\), then \(a-b=\cdots\)

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

Problem 34

In positive exponent and radical form, \(\dfrac{x^{-1}-y^{-1}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}=\cdots\)

A. \(\frac{\sqrt{x}-\sqrt{y}}{xy}\) B. \(\frac{\sqrt{y}-\sqrt{x}}{xy}\) C. \(\frac{\sqrt{x}+\sqrt{y}}{xy}\) D. \(xy(\sqrt{x}+\sqrt{y})\) E. \(xy(\sqrt{x}-\sqrt{y})\)

Problem 35

\(\dfrac{\sqrt[3]{2}+1}{\sqrt[3]{4}-\sqrt[3]{2}+1}\) can be simplified to...

A. \(\frac{1}{3}(\sqrt[3]{4}+2\sqrt[3]{2}+1)\) B. \(\frac{1}{3}(\sqrt[3]{4}+2\sqrt[3]{2}+1)\) C. \(\frac{1}{3}(\sqrt[3]{4}-2\sqrt[3]{2}+1)\) D. \(\frac{1}{3}(\sqrt[3]{4}-\sqrt[3]{2}+1)\) E. \(\frac{1}{3}(\sqrt[3]{4}+1)\)

Problem 36

\(\dfrac{(9+\sqrt{5})(2\sqrt{5}+1)}{\sqrt{5}+1}=\cdots\)

A. \(21\sqrt{5}\) B. \(19\) C. \(8\sqrt{5}\) D. \(15\) E. \(5\sqrt{5}\)

Problem 37

If positive integers \(a\) and \(b\) satisfy \(\sqrt{17+4\sqrt{15}}=a\sqrt{3}+b\sqrt{5}\), then \(b-a=\cdots\)

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

Problem 38

If integers \(a\) and \(b\) satisfy \(\dfrac{\sqrt{10}-\sqrt{5}}{\sqrt{10}+\sqrt{5}}=a+b\sqrt{2}\), then \(a+b=\cdots\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(5\)

Problem 39

If \(\dfrac{\frac{1}{2}-\frac{1}{\sqrt{5}}}{\frac{1}{2}+\frac{1}{\sqrt{5}}}=a+b\sqrt{5}\), then \(a+b=\cdots\)

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

Problem 40

If \(a=\dfrac{2+\sqrt{3}}{2-\sqrt{3}}\) and \(b=\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\), then \(a+b=\cdots\)

A. \(0\) B. \(1\) C. \(8\) D. \(10\) E. \(14\)

Problem 41

If \(p=(3+2\sqrt{2})^{-1}\) and \(q=(3-2\sqrt{2})^{-1}\), then \((1+p)^{-1}+(1+q)^{-1}=\cdots\)

A. \(1\) B. \(2\) C. \(4\) D. \(6\) E. \(8\)

Problem 42

Simplified form of \(\dfrac{(\sqrt{3}+\sqrt{7})(\sqrt{3}-\sqrt{7})}{2\sqrt{5}-4\sqrt{2}}\) is...

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

Problem 43

Simplified form of \(\dfrac{3\sqrt{3}}{\sqrt{2}+\sqrt{7}}\) is...

A. \(-\frac{3}{5}\sqrt{21}-\sqrt{6}\) B. \(-\frac{3}{5}\sqrt{21}+\sqrt{6}\) C. \(\frac{3}{5}\sqrt{20}-\frac{3}{5}\sqrt{5}\) D. \(\frac{3}{5}\sqrt{21}-\frac{3}{5}\sqrt{6}\) E. \(\frac{3}{5}\sqrt{21}+\frac{3}{5}\sqrt{6}\)

Problem 44

Simplified form of \(\dfrac{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}{\sqrt{3}+2}\) is...

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