# Square Roots and Cube Roots

1. By what least number 675 be multiplied to obtain a number which is a perfect cube ?
a. 5
b. 6
c. 7
d. 8
Correct Option: A
Explanation:
$675 = 5 \times 5 \times 3 \times 3 \times 3$ . To make it a perfect cube it must be multiplied by 5.

2. The largest four-digit number which is a perfect cube is
a. 9999
b. 9261
c. 8000
d. None
Correct Option: B
Explanation:
Clearly 9261 is a perfect cube satisfying the given property.

3. ${}^3\sqrt {\sqrt {0.000064} } = ?$
a. 0.02
b. 0.2
c. 2
d. None
Correct Option: B
Explanation:
$\sqrt {.000064} = \sqrt {\displaystyle\frac{{64}}{{{{10}^6}}}} = \displaystyle\frac{8}{{{{10}^3}}} = \displaystyle\frac{8}{{1000}} = .008$
$^3\sqrt {\sqrt {0.000064} } = 3\sqrt {.008}$
$^{{}^3\sqrt {\displaystyle\frac{8}{{1000}}} = \displaystyle\frac{2}{{10}} = 0.2}$

4. If 2*3 = $\sqrt {13}$ and 3*4=25, then the value of 5*12 is
a. $\sqrt {17}$
b. $\sqrt {29}$
c. 12
d. 13
Correct Option: D
Explanation
a*b = $\sqrt {{a^2} + {b^2}}$, So, 5*12 = $\sqrt {{5^2} + {{(12)}^2}} = \sqrt {25 + 144} = \sqrt {169} = 13$

5. If x * y * z = $\sqrt {\displaystyle\frac{{(x + 2)(y + 3)}}{{(z + 1)}}}$ , then (6*15*3) is
a. 2
b. 3
c. 4
d. None
Correct Option: D
Explanation:
(6*15*3) = $\sqrt {\displaystyle\frac{{(6 + 2)(15 + 3)}}{{(3 + 1)}}} =$ $\sqrt {\displaystyle\frac{{8 \times 18}}{4}} = \sqrt {36} = 6$

6. $\sqrt {12 + \sqrt {12 + \sqrt {12 +...... } } }$  = ?
a. 3
b. 4
c. 6
d. >6
Correct Option: B
Explanation:
In the given expression, let the value be x.
$\sqrt {12 + \sqrt {12 + \sqrt {12 +...... } } }$  = x
Squaring both sides,
$12 + \sqrt {12 + \sqrt {12 + \sqrt {12 +... } } } = x^2$
$\Rightarrow12 + x = x^2$
${x^2} - x - 12 = 0$ or (x-4) (x+3) = 0
or x = 4

7. The least square number which is exactly divisible by 10,12,15 and 18 is
a. 360
b. 400
c. 900
d. 1600
Correct Option: C
Explanation:
LCM of 10,12,15,18 = 180
= $2 \times 2 \times 3 \times 3 \times 5 = {2^2} \times {3^2} \times 5$
To make it a perfect square it must be multiplied by 5.
Required number = ${({2^2} \times {3^2} \times {5^2})}$=900

8. $\left( {\displaystyle\frac{{\sqrt 7 + \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }}} \right)$ is equal to
a. 6+$\sqrt {35}$
b. 6 - $\sqrt {35}$
c. 2
d. 1
Correct Option: A
Explanation:
$\displaystyle\frac{{\sqrt 7 + \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }} = \left( {\displaystyle\frac{{\sqrt 7 + \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }}} \right) \times \displaystyle\frac{{\sqrt 7 + \sqrt 5 }}{{\sqrt 7 + \sqrt 5 }}$ = ${\left( {\displaystyle\frac{{\sqrt 7 + \sqrt 5 }}{{(7 - 5)}}} \right)^2}$
= $\displaystyle\frac{{7 + 5 + 2\sqrt 7 \times \sqrt 5 }}{2}$
= $\displaystyle\frac{{12 + 2\sqrt {35} }}{2} = \left( {6 + \sqrt {35} } \right)$

9. $\dfrac{{\dfrac{1}{{\sqrt 9 }} - \dfrac{1}{{\sqrt {11} }}}}{{\dfrac{1}{{\sqrt 9 }} + \dfrac{1}{{\sqrt {11} }}}} \times \dfrac{{10 + \sqrt {99} }}{?} = \dfrac{1}{2}$
a. 2
b. 3
c. 4
d.None
Correct Option: A
Explanation:
Let $\left[ {\displaystyle\frac{{\displaystyle\frac{1}{{\sqrt 9 }} - \displaystyle\frac{1}{{\sqrt {11} }}}}{{\displaystyle\frac{1}{{\sqrt 9 }} + \displaystyle\frac{1}{{\sqrt {11} }}}}} \right] \times \displaystyle\frac{{10 + \sqrt {99} }}{x} = \displaystyle\frac{1}{2}$ . Then,
x = $\dfrac{{\dfrac{{\sqrt {11} - \sqrt 9 }}{{\sqrt {11} \times \sqrt 9 }}}}{{\dfrac{{\sqrt {11} + \sqrt 9 }}{{\sqrt {11} \times \sqrt 9 }}}} \times \left( {10 + \sqrt {99} } \right) \times 2$
x = $\dfrac{{\sqrt {11} - \sqrt 9 }}{{\sqrt {11} + \sqrt 9 }} \times \left( {10 + \sqrt {99} } \right) \times 2$
x = $\dfrac{{\sqrt {11} - \sqrt 9 }}{{\sqrt {11} + \sqrt 9 }} \times \dfrac{{\sqrt {11} - \sqrt 9 }}{{\sqrt {11} - \sqrt 9 }} \times \left( {10 + \sqrt {99} } \right) \times 2$
x = ${\left( {\displaystyle\frac{{\sqrt {11} - \sqrt 9 }}{{11 - 9}}} \right)^2} \times \left( {10 + \sqrt {99} } \right) \times 2$
x = $\left( {11 + 9 - 2\sqrt {99} } \right)\left( {10 + \sqrt {99} } \right)$
x =2 $\left( {10 - \sqrt {99} } \right)\left( {10 + \sqrt {99} } \right)$=2(100-99)=2

10. Which one of the following numbers has rational square root?
a.0.4
b. 0.09
c. 0.9
d. 0.025
Correct Option: B
Explanation:
$\sqrt {0.09} = \sqrt {\displaystyle\frac{9}{{100}}} = \displaystyle\frac{3}{{10}} = 0.3$ , which is rational
0.09 has rational square root.

11.The value of $\sqrt 2$ upto three places of decimal is
a. 1.410
b. 1.412
c. 1.413
d. 1.414
Correct Option: D
Explanation:
$\sqrt 2 = 1.414$

12.The square root of $\left( {8 + 2\sqrt 15 } \right)$ is
a. $\left( {\sqrt 2 + \sqrt 6 } \right)$
b. $\left( {\sqrt 5 + \sqrt 3 } \right)$
c. $\left( {2\sqrt 3 + 5\sqrt 5 } \right)$
d. ${2 + \sqrt 6 }$
Correct Option: B
Explanation:
$\left( {8 + 2\sqrt {15} } \right) = \left[ {{{\left( {\sqrt 5 } \right)}^2} + {{\left( {\sqrt 3 } \right)}^2} + 2 \times \sqrt 5 \times \sqrt 3 } \right]$ = ${\left( {\sqrt 5 + \sqrt 3 } \right)^2}$
$\sqrt {8 + 2\sqrt {15} } = \left( {\sqrt 5 + \sqrt 3 } \right)$

13. If ${\sqrt 6 }$ = 2.449, then the value of $\displaystyle\frac{{3\sqrt 2 }}{{2\sqrt 3 }}$ is
a. 0.6122
b. 1.223
c. 1.2245
d. 0.8163
Correct Option: C
Explanation:
$\dfrac{{3\sqrt 2 }}{{2\sqrt 3 }} = \dfrac{{3\sqrt 2 }}{{2\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} = \dfrac{{3\sqrt 6 }}{{2 \times 3}} = \dfrac{{\sqrt 6 }}{2}$
= $\displaystyle\frac{{2.449}}{2} = 1.2245$

14. $\displaystyle\frac{1}{{\left( {\sqrt 9 - \sqrt 8 } \right)}}$ - $\displaystyle\frac{1}{{\left( {\sqrt 8 - \sqrt 7 } \right)}}$ + $\displaystyle\frac{1}{{\left( {\sqrt 7 - \sqrt 6 } \right)}}$ - $\displaystyle\frac{1}{{\left( {\sqrt 6 - \sqrt 5 } \right)}}$ + $\displaystyle\frac{1}{{\left( {\sqrt 5 - \sqrt 4 } \right)}}$ = ?
a. 0
b. 1
c. 5
d. 1/3
Correct Option: C
Explanation:
$\displaystyle\frac{1}{{\sqrt 9 - \sqrt 8 }} = \displaystyle\frac{1}{{\sqrt 9 - \sqrt 8 }} \times \displaystyle\frac{{\sqrt 9 + \sqrt 8 }}{{\sqrt 9 + \sqrt 8 }}$
= $\displaystyle\frac{{\sqrt 9 + \sqrt 8 }}{{(9 - 8)}} = \left( {\sqrt 9 + \sqrt 8 } \right)$
similarly $\displaystyle\frac{1}{{\sqrt 8 - \sqrt 7 }} = \left( {\sqrt 8 + \sqrt 7 } \right)$
$\displaystyle\frac{1}{{\sqrt 7 - \sqrt 6 }} = \left( {\sqrt 7 + \sqrt 6 } \right)$
and $\displaystyle\frac{1}{{\sqrt 5 - \sqrt 4 }} = \left( {\sqrt 5 + \sqrt 4 } \right)$
Given Exp.
=$\left( {\sqrt 9 + \sqrt 8 } \right) - $$\left( {\sqrt 8 + \sqrt 7 } \right) + \left( {\sqrt 7 + \sqrt 6 } \right) -$$\left( {\sqrt 6 + \sqrt 5 } \right)$ + $\left( {\sqrt 5 + \sqrt 4 } \right)$
=$\left( {\sqrt 9 + \sqrt 4 } \right) = (3 + 2) = 5$

15. If ${\sqrt 2 }$ =1.4142, the square root of $\displaystyle\frac{{\left( {\sqrt 2 - 1} \right)}}{{\left( {\sqrt 2 + 1} \right)}}$ is equal to
a. 0.732
b. 0.3652
c. 1.3142
d. 0.4142
Correct Option: D
Explanation:
$\displaystyle\frac{{\sqrt 2 - 1}}{{\sqrt 2 + 1}}$ = $\displaystyle\frac{{\left( {\sqrt 2 - 1} \right)}}{{\left( {\sqrt 2 + 1} \right)}} \times \displaystyle\frac{{\left( {\sqrt 2 - 1} \right)}}{{\left( {\sqrt 2 - 1} \right)}}$ = ${\left( {\sqrt 2 - 1} \right)^2}$
$\sqrt {\left[ {\displaystyle\frac{{\sqrt 2 - 1}}{{\sqrt 2 + 1}}} \right]} = \left( {\sqrt 2 - 1} \right) = (1.4142 - 1)$
= 0.4142

16. What is the smallest number to be subtracted from 549162 in order to make it a perfect square ?
a. 28
b. 36
c. 62
d. 81
Correct Option: D
Explanation:

So 81 Should be subtracted to make the given number a perfect square.

17. If a $\displaystyle\frac{{\sqrt 5 + 1}}{{\sqrt 5 - 1}}$ and b = $\displaystyle\frac{{\sqrt 5 - 1}}{{\sqrt 5 + 1}}$, the value of $\left( {\displaystyle\frac{{{a^2} + ab + {b^2}}}{{{a^2} - ab + {b^2}}}} \right) =$
a. $\displaystyle\frac{3}{4}$
b. $\displaystyle\frac{4}{3}$
c. $\displaystyle\frac{3}{5}$
d. $\displaystyle\frac{5}{3}$
Correct Option: B
Explanation:
a = $\displaystyle\frac{{\sqrt 5 + 1}}{{\sqrt 5 - 1}} \times \displaystyle\frac{{\sqrt 5 + 1}}{{\sqrt 5 + 1}} = {\displaystyle\frac{{\left( {\sqrt 5 + 1} \right)}}{{(5 - 1)}}^2}$ =$\displaystyle\frac{{5 + 1 - 2\sqrt 5 }}{4} = \left[ {\displaystyle\frac{{3 - \sqrt 5 }}{2}} \right]$
b = $\displaystyle\frac{{\sqrt 5 - 1}}{{\sqrt 5 + 1}} \times \displaystyle\frac{{\sqrt 5 - 1}}{{\sqrt 5 - 1}} = {\displaystyle\frac{{\left( {\sqrt 5 - 1} \right)}}{{(5 - 1)}}^2}$ =$\displaystyle\frac{{5 + 1 - 2\sqrt 5 }}{4} = \left[ {\displaystyle\frac{{3 - \sqrt 5 }}{2}} \right]$
${a^2} + {b^2} = \displaystyle\frac{{{{\left( {3 + \sqrt 5 } \right)}^2}}}{4} + \displaystyle\frac{{{{\left( {3 - \sqrt 5 } \right)}^2}}}{4}$ =$\displaystyle\frac{{{{\left( {3 + \sqrt 5 } \right)}^2}}}{4} + \displaystyle\frac{{{{\left( {3 - \sqrt 5 } \right)}^2}}}{4}$
=$\displaystyle\frac{{2(9 + 5)}}{4} = 7$
Also, ab = $\displaystyle\frac{{(3 + \sqrt 5 )}}{2}$ $\displaystyle\frac{{3 - \sqrt 5 }}{2} = \displaystyle\frac{{(9 - 5)}}{4} = 1$
$\displaystyle\frac{{{a^2} + ab + {b^2}}}{{{a^2} - ab + {b^2}}} = \displaystyle\frac{{{a^2} + {b^2} + ab}}{{({a^2} + {b^2}) - ab}} = \displaystyle\frac{{7 + 1}}{{7 - 1}}$
=$\displaystyle\frac{8}{6} = \displaystyle\frac{4}{3}$

18. The expression $\left( {2 + \sqrt 2 } \right) + \dfrac{1}{{\left( {2 + \sqrt 2 } \right)}}{\rm{ }} - {\rm{ }}\dfrac{1}{{\left( {2 - \sqrt 2 } \right)}} = ?$
a. 2
b. $2\sqrt 2$
c. ${2 - \sqrt 2 }$
d. ${2 + \sqrt 2 }$
Correct Option: A
Explanation:
Given Expression = $\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{1}{{\left( {2 + \sqrt 2 } \right)}} \times \displaystyle\frac{{\left( {2 - \sqrt 2 } \right)}}{{\left( {2 - \sqrt 2 } \right)}}$ — $\displaystyle\frac{1}{{\left( {2 - \sqrt 2 } \right)}} \times \displaystyle\frac{{\left( {2 + \sqrt 2 } \right)}}{{\left( {2 + \sqrt 2 } \right)}}$
=$\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{{\left( {2 - \sqrt 2 } \right)}}{{\left( {4 - 2} \right)}} - \displaystyle\frac{{\left( {2 + \sqrt 2 } \right)}}{{\left( {4 - 2} \right)}}$
=$\left( {2 + \sqrt 2 } \right) + \displaystyle\frac{1}{2}\left( {2 - \sqrt 2 } \right) - \displaystyle\frac{1}{2}\left( {2 + \sqrt 2 } \right)$
$= \left( {2 + \sqrt 2 } \right) + 1 - \dfrac{{\sqrt 2 }}{2} - 1 - \dfrac{{\sqrt 2 }}{2} = 2$

19. $\sqrt {\displaystyle\frac{{0.081 \times 0.324 \times 4.624}}{{15.625 \times 0.0289 \times 72.9 \times 6.4}}} = ?$
a. 24
b. 2.4
c. 0.024
d. None
Correct Option: C
Explanation:
Total digits after the decimal points in the numerator = 9, as is denominator. So multiplying numerator and denominator by ${10}^9$ we get,
$\sqrt {\displaystyle\frac{{81 \times 324 \times 4624}}{{15625 \times 289 \times 729 \times 64}}}$ = $\left[ {\displaystyle\frac{{9 \times 18 \times 68}}{{125 \times 17 \times 27 \times 8}}} \right] = \displaystyle\frac{3}{{125}} = 0.024$

20. $\sqrt {\displaystyle\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}}$ is equal to
a. 9
b. 0.9
c. 99
d. 0.99
Correct Option: D
Explanation:
Sum of decimal places in the numerator and denominator under the radical sign being the same, we remove the decimal places:
Given Exp. = $\sqrt {\displaystyle\frac{{81 \times 484}}{{64 \times 625}}}$ = $\displaystyle\frac{{9 \times 22}}{{8 \times 25}} = 0.99$

21. If $\displaystyle\frac{x}{{\sqrt {2.25} }} = 550$, then the value of x is
a. 825
b. 82.5
c. 3666.66
d. 2
Correct Option: A
Explanation :
$\displaystyle\frac{x}{{\sqrt {2.25} }} = 550 \Rightarrow \displaystyle\frac{x}{{1.5}} = 550 \Rightarrow$ x = $(550 \times 1.5) \Rightarrow$ x = 825

22. $\sqrt {81} + \sqrt {0.81} = 10.09 - ?$
a. 1.19
b. 0.19
c. 1
d. 0.19
Correct Option: D
Explanation :
Let $\sqrt {81} + \sqrt {0.81}$  = 10.09 - $x$ $\Rightarrow$ $\sqrt {81} + \sqrt {\displaystyle\frac{{81}}{{100}}}$  = 10.09 - $x$
x = 10.09- (9 + 0.9) = 0.19

23. $\sqrt {\displaystyle\frac{{32.4}}{?}} = 2$
a. 9
b. 0.9
c. 0.09
d. None
Correct Option: D
Explanation:
Let $\sqrt {\displaystyle\frac{{32.4}}{x}} = 2$ Then $\displaystyle\frac{{32.4}}{x} = 4$
or 4x = 32.4 or x = 8.1

24. $\displaystyle\frac{{\sqrt {32} + \sqrt {48} }}{{\sqrt 8 + \sqrt {12} }} = ?$
a. $\sqrt 2$
b. 2
c. 4
d. 8
Correct Option: B
Explanation:
$\displaystyle\frac{{\sqrt {32} + \sqrt {48} }}{{\sqrt 8 + \sqrt {12} }} = \displaystyle\frac{{\sqrt {16 \times 2} + \sqrt {16 \times 3} }}{{\sqrt {4 \times 2} + \sqrt {4 \times 3} }}$ = $\displaystyle\frac{{4\left( {\sqrt 2 + \sqrt 3 } \right)}}{{2\left( {\sqrt 2 + \sqrt 3 } \right)}} = 2$

25. If $\sqrt {1 + \displaystyle\frac{x}{{144}}} = \displaystyle\frac{{13}}{{12}}$ then 'x' is equal to
a. 1
b. 12
c. 13
d. 25
Correct Option: D
Explanation:
$\sqrt {1 + \displaystyle\frac{x}{{144}}} = \displaystyle\frac{{13}}{{12}} \Rightarrow 1 + \displaystyle\frac{x}{{144}} = \displaystyle\frac{{169}}{{144}}$
$\displaystyle\frac{x}{{144}} = \left[ {\displaystyle\frac{{169}}{{144}} - 1} \right]$ or $\displaystyle\frac{x}{{144}}$ = $\displaystyle\frac{{25}}{{144}}$ or x = 25