# Power, Indices and Surds -Quantitative Aptitude

Question 01
\left(\frac{8}{125}\right)^{\frac{-4}{3}} \ simplifies \ to: \\\ \\ (a) \frac{625}{16}\\\ \\ (b) \frac{625}{8} \\\ \\ (c) \frac{625}{32} \\\ \\ (d) \frac{16}{625} \\\ \\

Read Solution
Sol:  \left(\frac{8}{125}\right)^{\frac{-4}{3}} \\\ \\ ⟹ \left(\frac{125}{8}\right)^{\frac{3}{4}} \\\ \\ ⟹  \left[\left(\frac{5}{2}\right)^{3}\right]^{\frac{4}{3}} \\\ \\ ⟹ \left(\frac{5}{2}\right)^{4} \\\ \\⟹  \frac{625}{16}

Option (a) is the right answer

Question 02
The \ value \ of \ ( 256)^{0.16} \times ( 16)^{0.18} is: \\ \\ (a) 4 \\ \\ (b) -4 \\ \\ (c) 16 \\ \\ (d) 256 \\ \\

Read Solution
( 256)^{0.16} \times ( 16)^{0.18} \\\ \\ ⟹ \ ( 4)^{4\times 0.16} \times ( 4)^{2\times 0.18} \\\ \\ ⟹ ( 4)^{0.64} \times ( 4)^{0.36} ⟹ ( 4)^{1} =4

Option (a) is the right answer

Question 03
Simplify: \\\ \\ \frac{0.41\times 0.41\times 0.41+0.69\times 0.69\times 0.69}{0.41\times 0.41-0.41\times 0.69+0.69\times 0.69} \\\ \\ (a) 0.28\\ \\ (b) 1.41 \\ \\ (c) 1.1 \\ \\ (d) 2.8 \\ \\

Read Solution
Sol: \frac{0.41\times 0.41\times 0.41+0.69\times 0.69\times 0.69}{0.41\times 0.41-0.41\times 0.69+0.69\times 0.69} \\\ \\ ⟹ \frac{( 0.41)^{3} +( 0.69)^{3}}{( 0.41)^{2} -0.41\times 0.69+( 0.69)^{2}} \\\ \\ ⟹ \frac{( 0.41+0.69)\left( 0.41^{2} +0.69^{2} -0.2829\right)}{0.1681-0.2829+0.4761} \\\ \\ ⟹ 0.41+0.69=1.10

option (c) is the right answer

Question 04
Which of the following number is the least?
( 0.5)^{2} ,\sqrt{0.49} ,\sqrt[3]{0.0008} ,0.23 \\\ \\ (a) ( 0.5)^{2} \\\ \\ (b) \sqrt{0.49} \\\ \\ (c) \sqrt[3]{0.0008} \\\ \\ (d) 0.23 \\ \\

Read Solution
\ \ \ ( 0.5)^{2} ,\sqrt{0.49} ,\sqrt[3]{0.0008} ,0.23 \\\ \\ ⟹ 0.25 , 0.7,\underline{0.2}, \ 0.23 \\\ \\ \sqrt[3]{0.0008} \ is \ least

option (c) is the right answer

Question 05
Which one is the greatest ?
\ \sqrt{2} ,\sqrt[3]{3} ,\sqrt[4]{5} ,\sqrt[3]{2} \\\ \\ (a) \sqrt[4]{5} \\ \\ (b) \sqrt{2} \\ \\ (c) \sqrt[3]{3} \\ \\ (d) \sqrt[3]{2}\\\ \\ Read Solution

\sqrt{2} ,\sqrt[3]{3} ,\sqrt[4]{5} ,\sqrt[3]{2} \\\ \\ ⟹ 2^{\frac{1}{2}} ,3^{\frac{1}{3}} ,5^{\frac{1}{4}} ,2^{\frac{1}{3}} \\\ \\ ⟹ 2^{\frac{6}{12}} ,3^{\frac{4}{12}} ,5^{\frac{3}{12}} ,2^{\frac{4}{12}} \\\ \\ ⟹ \sqrt[12]{2^{6}} ,\sqrt[12]{3^{4}} ,\sqrt[12]{5^{3}} ,\sqrt[12]{2^{4}} \\\ \\ ⟹ \sqrt[12]{64} ,\sqrt[12]{81} ,\sqrt[12]{125} ,\sqrt[12]{16} \\\ \\ \sqrt[12]{125} \ means \ \sqrt[4]{5} \ is \ greatest.

Option (a) is the right answer

Question 06
Find the value of :
\sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{7+4\sqrt{3}}}} \\\ \\ (a) 1 \\ \\ (b) 2 \\ \\ (c) 3 \\ \\  (d) 8 \\\ \\ Read Solution

\sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{7+4\sqrt{3}}}} \\\ \\ ⟹ \sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{4+3+2\times 2\sqrt{3}}}} \\\ \\ ⟹ \sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{\left( 2+\sqrt{3}\right)^{2}}}} \\\ \\ ⟹ \sqrt{-\sqrt{3} +\sqrt{3+8\left( 2+\sqrt{3}\right)}} \\\ \\ ⟹ \sqrt{-\sqrt{3+}\sqrt{3+16+8\sqrt{3}}} \\\ \\ ⟹ \sqrt{-\sqrt{3+}\sqrt{\left(\sqrt{3}\right)^{2} +( 4)^{2} +2\ \times 4\ \times \sqrt{3}}} \\\ \\ ⟹ \sqrt{-\sqrt{3} +\sqrt{\left( 4+\sqrt{3}\right)^{2}}} \\\ \\ ⟹ \sqrt{-\sqrt{3} +4+\sqrt{3}} \\\ \\ ⟹  \sqrt{4}=2 \\\ \\

Option (b) is the right answer

Question 07
\sqrt{\sqrt[3]{0.004096}}\ is \ equal \ to \\\ \\ (a) 4 \\ \\ (b) 0.4 \\ \\ (c) 0.04 \\ \\  (d) 0.004 \\\ \\ Read Solution

\sqrt{\sqrt[3]{0.004096}} \\\ \\ ⟹ \sqrt{0.16} \ \ \ \ [16 ^{3}=4096] \\\ \\ ⟹ \sqrt{0.4\times 0.4} \\\ \\ ⟹ 0.4

Option (b) is the right answer

Question 08
Find the value of:
\sqrt{8} +2\sqrt{32} -3\sqrt{18} +4\sqrt{50} \\\ \\ (a) 8.789 \\ \\ (b) 8.526 \\ \\ (c) 8.426 \\ \\  (d) 8.484 \\\ \\ Read Solution

\sqrt{8} +2\sqrt{32} -3\sqrt{18} +4\sqrt{50} \\\ \\ ⟹ 2\sqrt{2} +8\sqrt{2} -24\sqrt{2} +20\sqrt{2} \\\ \\ ⟹ 6\sqrt{2} \\\ \\ ⟹ 6 \times 1.414=8.484 \\\ \\ \\\ \\

Option (d) is the right answer

Question 09
If \sqrt{15} =3.88, \\ \\ then \ what \ is \ the \ value \ of \\ \\ \sqrt{\frac{5}{3}} \\\ \\ (a) 1.273 \\ \\ (b) 1.2934 \\ \\ (c) 1.2933 \\ \\  (d) 1.295 \\\ \\ Read Solution

\sqrt{\frac{5}{3}}= \sqrt{\frac{5\times 3}{3\times 3}}= \sqrt{\frac{15}{9}} \\\ \\ ⟹ \frac{\sqrt{15}}{3}= \frac{3.88}{3}=1.2933

Option (c) is the right answer

Question 10
Find the value of
\left( 3+2\sqrt{2}\right)^{-3} +\left( 3-2\sqrt{2}\right)^{-3} is: \\\ \\ (a) 189 \\ \\ (b) 180 \\ \\ (c) 108 \\ \\ (d) 198 \\\ \\ Read Solution

Sol: \left( 3+2\sqrt{2}\right)^{-3} +\left( 3-2\sqrt{2}\right)^{-3} \\\ \\\ ⟹ \left(\frac{1}{3+2\sqrt{2}}\right)^{3} +\left(\frac{1}{3-2\sqrt{2}}\right)^{3} \\\ \\ ⟹ \left(\frac{1}{3+2\sqrt{2}} \times \frac{3-2\sqrt{2}}{3-2\sqrt{2}}\right)^{3} +\left(\frac{1}{3-2\sqrt{2}} \times \frac{3+2\sqrt{2}}{3+2\sqrt{2}}\right)^{3} \\\ \\ ⟹ \left(\frac{3-2\sqrt{2}}{9-8}\right)^{3} +\left(\frac{3+2\sqrt{2}}{9-8}\right)^{3} \\\ \\ ⟹ \left( 3-2\sqrt{2}\right)^{3} +\left( 3+2\sqrt{2}\right)^{3} \\\ \\ ⟹ \left( 3-2\sqrt{2} +3+2\sqrt{2}\right)\left[\left( 3-2\sqrt{2}\right)^{2} +\left( 3+2\sqrt{2}\right)^{2} -\left( 3-2\sqrt{2}\right)\left( 3+2\sqrt{2}\right)\right] \\\ \\ ⟹ ( 6)( 17+17-1) \\\ \\ ⟹ (6)(33) \\\ \\ ⟹ 198 \\\ \\

Option (d) is the right answer

You cannot copy content of this page