Square Roots and Cube Roots

\text{1. Evaluate} \,\, \sqrt{78 + \sqrt{5 + 4}}

 

A. 4

B. 8

C. 9

D. 7

Ans: C

Explanation:

\sqrt{78 + \sqrt{5 +4}} = \sqrt{78 + \sqrt{9}} = \sqrt{78 + 3} = \sqrt{81} = 9

 

 

 

\text{2. find the value of } \sqrt{1\frac{9}{16}}

 

 

\text{A. } 1\frac{1}{2}

 

\text{B. } 1\frac{1}{4}

 

\text{C. } 2\frac{1}{2}

 

\text{D. } \frac{4}{5}

 

 

Ans: B

Explanation:

\sqrt{1\frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} = 1\frac{1}{4}

 

 

 

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

 

 

A. 4

B. 6

C. 8

D. 12

Ans: B

Explanation:

\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}} \,\, = \,\, \sqrt{25+\sqrt{108+\sqrt{154+15}}}

 

 

= \sqrt{25+\sqrt{108+\sqrt{169}}}

 

 

= \sqrt{25+\sqrt{108+{13}}}

 

 

= \sqrt{25+\sqrt{121}}

 

 

= \sqrt{25 + 11}

 

= \sqrt{36}

 

= 6

 

 

\text{4. Evaluate } \sqrt{41-\sqrt{21+\sqrt{19-\sqrt{9}}}}

 

 

A. 4

B. 6

C. 8

D. 2

Ans: B

Explanation:

\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}} \,\, = \,\, \sqrt{41 - \sqrt{21 + \sqrt{19 - {3}}}}

 

 

= \sqrt{41 - \sqrt{21 + \sqrt{16}}}

 

 

= \sqrt{41 - \sqrt{21 + 4}}

 

 

= \sqrt{41 - \sqrt{25}}

 

 

= \sqrt{41 - 5}

 

 

= \sqrt{36}

 

 

= 6

 

 

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

 

 

A. 5

B. 6

C. 8

D. 9

Ans: A

Explanation:

\frac{\sqrt{625}}{11} \times \frac{14}{\sqrt{25}} \times \frac{11}{\sqrt{196}} = \frac{25}{11} \times \frac{14}{5} \times \frac{11}{14} = 5

 

 

 

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

 

A. 6

B. 8

C. 12

D. 16

Ans: C

Explanation:

\sqrt{3^n} = 729

 

\sqrt{3^n} = 3^6

 

\text{squaring on both sides}

 

\left ( \sqrt{3^n} \right )^2 = \left ( 3^6 \right )^2

 

3^n = 3^{12}

 

\therefore \text{n} = 12

 

 

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

 

A. 18

B. 24

C. 28

D. 32

Ans: C

Explanation:

\sqrt{18 \times 14 \times x} = 84

 

\text{Squaring on both sides}

 

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

 

18 \times 14 \times x = 84 \times 84

 

x = \frac{84 \times 84}{18 \times 14} = 28

 

 

 

\text{8. find the value of} \, \sqrt{6+\sqrt{6+\sqrt{6+....}}}

 

 

A. 1

B. 2

C. 3

D. 4

Ans: C

Explanation:

\text{Let} \, x \, \text{be the value of given expression}

 

x = \sqrt{6+\sqrt{6+\sqrt{6+....}}}

 

 

\text{Squaring on both sides}

 

x^2 = 6+ \sqrt{6+\sqrt{6+\sqrt{6+....}}}

 

 

x^2 = 6+x

 

x^2 - x- 6 = 0

 

x^2 - 3x + 2x - 6 = 0

 

x\left ( x - 3 \right ) + 2 \left ( x - 3 \right ) = 0

 

\left ( x + 2 \right ) \left ( x - 3 \right ) = 0

 

x = -\text{2 or 3}

 

\text{Since } x \, \text{value can't be negative}

 

\text{So } x = 3

 

 

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

 

 

A. 4.262

B. 3.121

C. 5.368

D. 5.828

Ans: D

Explanation:

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

 

 

\text{To rationalize multiply it with} \,\, \frac{\sqrt{2}+ 1}{\sqrt{2}+1}

 

 

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

 

 

= \frac{2 + 1 + 2\sqrt{2}}{2 - 1}

 

 

= \frac{3 + 2\sqrt{2}}{1}

 

 

= 3 + 2\sqrt{2}

 

= 3 + 2 \left ( 1.414 \right )

 

= 3 + 2.828

 

= 5.828

 

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

 

 

A. 6

B. 8.1

C. 9

D. 9.8

Ans: B

Explanation:

\text{Let} \, \sqrt{\frac{32.4}{x}} = 2

 

 

\text{Squarng on both sides}

 

\frac{32.4}{x} = 4

 

 

x = \frac{32.4}{4}

 

 

x = 8.1