# Square Roots and Cube Roots - 2

9. $\displaystyle\frac{{\displaystyle\frac{1}{{\sqrt 9 }} - \displaystyle\frac{1}{{\sqrt {11} }}}}{{\displaystyle\frac{1}{{\sqrt 9 }} + \displaystyle\frac{1}{{\sqrt {11} }}}} + \displaystyle\frac{{10 + \sqrt {99} }}{?} = \displaystyle\frac{1}{2}$
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 = $\displaystyle\frac{{\sqrt {11} - \sqrt 9 }}{{\sqrt {11} + \sqrt 9 }} \times \displaystyle\frac{{\sqrt {11} - \sqrt 9 }}{{\sqrt {11} - \sqrt 9 }} \times \left( {10 + \sqrt {99} } \right) \times 2$

= ${\left( {\displaystyle\frac{{\sqrt {11} - \sqrt 9 }}{{11 - 9}}} \right)^2} \times \left( {10 + \sqrt {99} } \right) \times 2$

= $\left( {11 + 9 - 2\sqrt {99} } \right)\left( {10 + \sqrt {99} } \right)$
=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 $\left( {8 + 2\sqrt 5 } \right)$ 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:
$\displaystyle\frac{{3\sqrt 2 }}{{2\sqrt 3 }} = \displaystyle\frac{{3\sqrt 2 }}{{2\sqrt 3 }} \times \displaystyle\frac{{\sqrt 3 }}{{\sqrt 3 }} \times \displaystyle\frac{{\sqrt 6 }}{2}$
= $\displaystyle\frac{{2449}}{2} = 1.2245$

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.