# Square Roots and Cube Roots

$\text{1.&space;Evaluate}&space;\,\,&space;\sqrt{78&space;+&space;\sqrt{5&space;+&space;4}}$

A. 4

B. 8

C. 9

D. 7

Ans: C

Explanation:

$\sqrt{78&space;+&space;\sqrt{5&space;+4}}&space;=&space;\sqrt{78&space;+&space;\sqrt{9}}&space;=&space;\sqrt{78&space;+&space;3}&space;=&space;\sqrt{81}&space;=&space;9$

$\text{2.&space;find&space;the&space;value&space;of&space;}&space;\sqrt{1\frac{9}{16}}$

$\text{A.&space;}&space;1\frac{1}{2}$

$\text{B.&space;}&space;1\frac{1}{4}$

$\text{C.&space;}&space;2\frac{1}{2}$

$\text{D.&space;}&space;\frac{4}{5}$

Ans: B

Explanation:

$\sqrt{1\frac{9}{16}}&space;=&space;\sqrt{\frac{25}{16}}&space;=&space;\frac{5}{4}&space;=&space;1\frac{1}{4}$

$\text{3.&space;The&space;value&space;of&space;}&space;\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}&space;\,\,\,&space;\text{is}$

A. 4

B. 6

C. 8

D. 12

Ans: B

Explanation:

$\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}&space;\,\,&space;=&space;\,\,&space;\sqrt{25+\sqrt{108+\sqrt{154+15}}}$

$=&space;\sqrt{25+\sqrt{108+\sqrt{169}}}$

$=&space;\sqrt{25+\sqrt{108+{13}}}$

$=&space;\sqrt{25+\sqrt{121}}$

$=&space;\sqrt{25&space;+&space;11}$

$=&space;\sqrt{36}$

$=&space;6$

$\text{4.&space;Evaluate&space;}&space;\sqrt{41-\sqrt{21+\sqrt{19-\sqrt{9}}}}$

A. 4

B. 6

C. 8

D. 2

Ans: B

Explanation:

$\sqrt{41&space;-&space;\sqrt{21&space;+&space;\sqrt{19&space;-&space;\sqrt{9}}}}&space;\,\,&space;=&space;\,\,&space;\sqrt{41&space;-&space;\sqrt{21&space;+&space;\sqrt{19&space;-&space;{3}}}}$

$=&space;\sqrt{41&space;-&space;\sqrt{21&space;+&space;\sqrt{16}}}$

$=&space;\sqrt{41&space;-&space;\sqrt{21&space;+&space;4}}$

$=&space;\sqrt{41&space;-&space;\sqrt{25}}$

$=&space;\sqrt{41&space;-&space;5}$

$=&space;\sqrt{36}$

$=&space;6$

$\text{5.&space;}&space;\frac{\sqrt{625}}{11}&space;\times&space;\frac{14}{\sqrt{25}}&space;\times&space;\frac{11}{\sqrt{196}}&space;\,\,&space;\text{equal&space;to}$

A. 5

B. 6

C. 8

D. 9

Ans: A

Explanation:

$\frac{\sqrt{625}}{11}&space;\times&space;\frac{14}{\sqrt{25}}&space;\times&space;\frac{11}{\sqrt{196}}&space;=&space;\frac{25}{11}&space;\times&space;\frac{14}{5}&space;\times&space;\frac{11}{14}&space;=&space;5$

$\text{6.&space;}&space;\sqrt{3^n}&space;=&space;729&space;\text&space;\,\,&space;\text{then&space;the&space;value&space;of&space;n&space;is}$

A. 6

B. 8

C. 12

D. 16

Ans: C

Explanation:

$\sqrt{3^n}&space;=&space;729$

$\sqrt{3^n}&space;=&space;3^6$

$\text{squaring&space;on&space;both&space;sides}$

$\left&space;(&space;\sqrt{3^n}&space;\right&space;)^2&space;=&space;\left&space;(&space;3^6&space;\right&space;)^2$

$3^n&space;=&space;3^{12}$

$\therefore&space;\text{n}&space;=&space;12$

$\text{7.&space;If&space;}&space;\,&space;\sqrt{18&space;\times&space;14&space;\times&space;x}&space;=&space;84.&space;\,&space;\text{Then&space;find&space;the&space;value&space;of&space;}&space;x.$

A. 18

B. 24

C. 28

D. 32

Ans: C

Explanation:

$\sqrt{18&space;\times&space;14&space;\times&space;x}&space;=&space;84$

$\text{Squaring&space;on&space;both&space;sides}$

$\left&space;(&space;\sqrt{18&space;\times&space;14&space;\times&space;x}&space;\right&space;)^2&space;=&space;\left&space;(&space;84&space;\right&space;)^2$

$18&space;\times&space;14&space;\times&space;x&space;=&space;84&space;\times&space;84$

$x&space;=&space;\frac{84&space;\times&space;84}{18&space;\times&space;14}&space;=&space;28$

$\text{8.&space;find&space;the&space;value&space;of}&space;\,&space;\sqrt{6+\sqrt{6+\sqrt{6+....}}}$

A. 1

B. 2

C. 3

D. 4

Ans: C

Explanation:

$\text{Let}&space;\,&space;x&space;\,&space;\text{be&space;the&space;value&space;of&space;given&space;expression}$

$x&space;=&space;\sqrt{6+\sqrt{6+\sqrt{6+....}}}$

$\text{Squaring&space;on&space;both&space;sides}$

$x^2&space;=&space;6+&space;\sqrt{6+\sqrt{6+\sqrt{6+....}}}$

$x^2&space;=&space;6+x$

$x^2&space;-&space;x-&space;6&space;=&space;0$

$x^2&space;-&space;3x&space;+&space;2x&space;-&space;6&space;=&space;0$

$x\left&space;(&space;x&space;-&space;3&space;\right&space;)&space;+&space;2&space;\left&space;(&space;x&space;-&space;3&space;\right&space;)&space;=&space;0$

$\left&space;(&space;x&space;+&space;2&space;\right&space;)&space;\left&space;(&space;x&space;-&space;3&space;\right&space;)&space;=&space;0$

$x&space;=&space;-\text{2&space;or&space;3}$

$\text{Since&space;}&space;x&space;\,&space;\text{value&space;can't&space;be&space;negative}$

$\text{So&space;}&space;x&space;=&space;3$

$\text{9.&space;If&space;}&space;\sqrt{2}&space;=&space;1.414&space;\text&space;\,&space;\text&space;{find&space;the&space;value&space;of&space;}&space;\,&space;\frac{\sqrt{2}&space;+&space;1}{\sqrt{2}&space;-&space;1}$

A. 4.262

B. 3.121

C. 5.368

D. 5.828

Ans: D

Explanation:

$\frac{\sqrt{2}&space;+&space;1}{\sqrt{2}&space;-&space;1}&space;\,\,&space;\text{in&space;this&space;fraction&space;denominator&space;is&space;irrational.}$

$\text{To&space;rationalize&space;multiply&space;it&space;with}&space;\,\,&space;\frac{\sqrt{2}+&space;1}{\sqrt{2}+1}$

$\frac{\sqrt{2}&space;+&space;1}{\sqrt{2}&space;-&space;1}&space;\times&space;\frac{\sqrt{2}&space;+&space;1}{\sqrt{2}&space;+&space;1}&space;=&space;\frac{\left&space;(&space;\sqrt{2}&space;+&space;1&space;\right&space;)^2}{\left&space;(&space;\sqrt{2}&space;\right&space;)^2&space;-&space;\left&space;(&space;1&space;\right&space;)^2}$

$=&space;\frac{2&space;+&space;1&space;+&space;2\sqrt{2}}{2&space;-&space;1}$

$=&space;\frac{3&space;+&space;2\sqrt{2}}{1}$

$=&space;3&space;+&space;2\sqrt{2}$

$=&space;3&space;+&space;2&space;\left&space;(&space;1.414&space;\right&space;)$

$=&space;3&space;+&space;2.828$

$=&space;5.828$

$\text{10.&space;What&space;will&space;come&space;in&space;the&space;place&space;of&space;question&space;mark}&space;\,\,&space;\sqrt{\frac{32.4}{?}}&space;=&space;2$

A. 6

B. 8.1

C. 9

D. 9.8

Ans: B

Explanation:

$\text{Let}&space;\,&space;\sqrt{\frac{32.4}{x}}&space;=&space;2$

$\text{Squarng&space;on&space;both&space;sides}$

$\frac{32.4}{x}&space;=&space;4$

$x&space;=&space;\frac{32.4}{4}$

$x&space;=&space;8.1$