# Power, Surds and Indices – Aptitude Problems and Answers

Question 01
Select the right option for value of “a”
a=\frac{1}{2-\sqrt{3}} +\frac{1}{3-\sqrt{8}} +\frac{1}{4-\sqrt{15}} \\\ \\ (a) a<18 \ but \ a \neq 9 \\ \\ (b) a>18 \\ \\ (c) a=18 \\ \\ (d) a=9 \\ \\ Read Solution

\frac{1}{2-\sqrt{3}} =\frac{1}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}} \\\ \\ ⟹ \frac{2+\sqrt{3}}{4-3} =2+\sqrt{3} \\\ \\ Similarly, \\\ \\ \frac{1}{3-\sqrt{8}}= 3+\sqrt{8} \\\ \\ \frac{1}{4-\sqrt{15}}= 4+\sqrt{15} \\\ \\ Now \ put \ the \ values \\\ \\ ⟹ 2+\sqrt{3} +3+\sqrt{8} +4+\sqrt{15} \\\ \\ ⟹ 9+\sqrt{3} +2\sqrt{2} +\sqrt{15} \\\ \\ a=9< 9+\sqrt{3} +2\sqrt{2} +\sqrt{15} < 18 \\ \\ \left(\sqrt{3} =1.73,\sqrt{2} =1.41,\sqrt{15} =3.9\right) \\\ \\ a=9< 9+1.73+( 2\times 1.41) +3.9< 18 \\\ \\ a=9< 17.4< 18 \\\ \\ So, a<18 \ but\ a \neq 9

Option (a) is the right answer

Question 02
2+\frac{6}{\sqrt{3}} +\frac{1}{2+\sqrt{3}} +\frac{1}{\sqrt{3} -2} \ equals\ to \\\ \\ (a) \left( 2\sqrt{3}\right) \\ \\ (b) -\left( 2\sqrt{3}\right) \\ \\ (c) 1 \\ \\ (d) 2 \\ \\ Read Solution

2+\frac{6}{\sqrt{3}} +\frac{1}{2+\sqrt{3}} +\frac{1}{\sqrt{3} -2} \\\ \\ ⟹ 2+\frac{2\times 3\sqrt{3}}{\sqrt{3} \times \sqrt{3}} +\frac{1}{2+\sqrt{3}} -\frac{1}{2-\sqrt{3}} \\\ \\ ⟹ 2+2\sqrt{3} +\left(\frac{\left( 2-\sqrt{3}\right) -\left( 2+\sqrt{3}\right)}{\left( 2-\sqrt{3}\right)\left( 2+\sqrt{3}\right)}\right) \\\ \\ ⟹ 2+2\sqrt{3} +\left(\frac{2-\sqrt{3} -2-\sqrt{3}}{4+2\sqrt{3} -2\sqrt{3} -3}\right) \\\ \\ ⟹ 2+2\sqrt{3} -2\sqrt{3} =2

Option (d) is the right answer

Question 03
Find the smallest number
2^{250} ,3^{150} ,5^{100} \ and\ 4^{200} \\\ \\ (a) 4^{200} \\ \\  (b) 5^{100} \\ \\ (c) 3^{150} \\ \\ (d) 2^{250} \\ \\ Read Solution

2^{250} , 3^{150} , 5^{100} , 4^{200} \\\ \\ ⟹ \left( 2^{5}\right)^{50} ,\left( 3^{3}\right)^{50} ,\left( 5^{2}\right)^{50} ,\left( 4^{4}\right)^{50} \\\ \\ ⟹ ( 32)^{50} ,( 9)^{50} ,( 25)^{50} ,( 256)^{50} \\\ \\ (25) ^{50} means\ 5 ^{100}\ is \ smallest.

Option (b) is the right answer

Question 04
If 5\sqrt{5} \times 5^{3} \div 5^{-\frac{3}{2}} =5^{a+2}, then\ the\ value\ of\ a\ is \\\ \\ (a) 4 \\ \\ (b) 5 \\ \\ (c) 6 \\ \\ (d) 8 \\ \\ Read Solution

5\sqrt{5} \times 5^{3} \div 5^{-\frac{3}{2}} =5^{a+2} \\ \\ ⟹ 5^{1} \times 5^{\frac{1}{2}} \times 5^{3} \div 5^{-\frac{3}{2}} =5^{a+2} \\ \\ ⟹ 5^{1+\frac{1}{2} +3-\left( -\frac{3}{2}\right)} =5^{a+2} \\ \\ ⟹ 5^{1+\frac{1}{2} +3+\frac{3}{2}} =5^{a+2} \\ \\ ⟹ 5^{\frac{4+2+12+6}{2}} =5^{a+2} \\ \\ ⟹ 5^{\frac{24}{4}} =5^{a+2} \\ \\ ⟹ 5^{6} =5^{a+2}

⟹ a+2=6
⟹ a=4

Option (a) is the right answer

Question 05
The value of
\frac{1}{1+\sqrt{2}} +\frac{1}{\sqrt{2} +\sqrt{3}} +\frac{1}{\sqrt{3} +\sqrt{4}}+\frac{1}{\sqrt{4} +\sqrt{5}} +\frac{1}{\sqrt{5} +\sqrt{6}} +\frac{1}{\sqrt{6} +\sqrt{7}} +\frac{1}{\sqrt{7} +\sqrt{8}} +\frac{1}{\sqrt{8} +\sqrt{9}} \ is \\\ \\ (a) 0 \\ \\ (b) 4 \\ \\ (c) 2 \\ \\ (d) 1 \\ \\ Read Solution

\frac{1}{1+\sqrt{2}} +\frac{1}{\sqrt{2} +\sqrt{3}} +\frac{1}{\sqrt{3} +\sqrt{4}} +\\ \\ \frac{1}{\sqrt{4} +\sqrt{5}} +\frac{1}{\sqrt{5} +\sqrt{6}} +\frac{1}{\sqrt{6} +\sqrt{7}} + \\ \\ \frac{1}{\sqrt{7} +\sqrt{8}} +\frac{1}{\sqrt{8} +\sqrt{9}} \\\ \\ ⟹ \frac{1}{\sqrt{2} +1} =\frac{1}{\sqrt{2} +1} \times \frac{\sqrt{2} -1}{\sqrt{2} -1} \\\ \\ ⟹ \frac{\sqrt{2} -1}{2-1} =\sqrt{2} -1 \\\ \\ Similarly, \frac{1}{\sqrt{2} +\sqrt{3}} =\sqrt{3} -\sqrt{2}\ and\ so\ on \\\ \\ Now\ put\ the\ values \\\ \\ \sqrt{2} -1+\sqrt{3} -\sqrt{2} +\sqrt{4} -\sqrt{3} +\\ \\ \sqrt{5} -\sqrt{4} +\sqrt{6} -\sqrt{5} +\sqrt{7} -\sqrt{6} +\\ \\ \sqrt{8} -\sqrt{7} +\sqrt{9} -\sqrt{8} \\\ \\ ⟹ \sqrt{9} -1 \\\ \\ ⟹ 3-1=2 \\ \\

Option (c) is the right answer