# Power, Indices and Surds – Solved MCQ

Question 01
\left(\frac{1}{1.4} +\frac{1}{4.7} +\frac{1}{7.10} +\frac{1}{10.13} +\frac{1}{13.16}\right)\ is\ equal\ to\\\ \\ (a) \frac{1}{3}           (b) \frac{5}{16}\\\ \\ (c) \frac{3}{8}           (d) \frac{41}{7280}\\\ \\ Read Solution

\left(\frac{1}{1.4} +\frac{1}{4.7} +\frac{1}{7.10} +\frac{1}{10.13} +\frac{1}{13.16}\right) \\\ \\ ⟹ \frac{1}{1\times 4} +\frac{1}{4\times 7} +\frac{1}{7\times 10} +\frac{1}{10\times 13} +\frac{1}{13\times 16} \\\ \\ Multiplying\ \& \ dividing\ the\ expression\ with\ 3,\ we\ get: \\\ \\ ⟹ \frac{3}{3}\left[\frac{1}{1\times 4} +\frac{1}{4\times 7} +\frac{1}{7\times 10} +\frac{1}{10\times 13} +\frac{1}{13\times 16}\right] \\\ \\ ⟹ \frac{1}{3}\left[\frac{3}{1\times 4} +\frac{3}{4\times 7} +\frac{3}{7\times 10} +\frac{3}{10\times 13} +\frac{3}{13\times 16}\right] \\\ \\ Rewriting\ the\ fractions\ inside\ the\ bracket,\ we\ get: \\\ \\ ⟹ \frac{1}{3}\left[\left(\frac{1}{1} -\frac{1}{4}\right) +\left(\frac{1}{4} -\frac{1}{7}\right) +\left(\frac{1}{7} -\frac{1}{10}\right) +\left(\frac{1}{10} -\frac{1}{13}\right) +\left(\frac{1}{13} -\frac{1}{16}\right)\right] \\\ \\ ⟹ \frac{1}{3}\left[\frac{1}{1} -\frac{1}{4} +\frac{1}{4} -\frac{1}{7} +\frac{1}{7} -\frac{1}{10} +\frac{1}{10} -\frac{1}{13} +\frac{1}{13} -\frac{1}{16}\right] \\\ \\ ⟹ \frac{1}{3}\left(\frac{1}{1} -\frac{1}{16}\right) \\\ \\ ⟹ \frac{1}{3}\left(\frac{16-1}{16}\right) \\\ \\ ⟹ \frac{1}{3}\left(\frac{15}{16}\right) \\\ \\ ⟹ \frac{5}{16}\\\ \\

Option (b) is the right answer

Question 02
The greatest among
\sqrt{7} -\sqrt{5} ,\sqrt{5} -\sqrt{3} ,\sqrt{9} -\sqrt{7} ,\sqrt{11} -\sqrt{9}\ is \\\ \\ Read Solution

\sqrt{7} -\sqrt{5} \\\ \\ ⟹ \ \left(\sqrt{7} -\sqrt{5}\right)\left(\frac{\sqrt{7} +\sqrt{5}}{\sqrt{7} +\sqrt{5}}\right) \\\ \\ ⟹ \frac{7+\sqrt{35} -\sqrt{35} -5}{\sqrt{7} +\sqrt{5}}= \frac{2}{\sqrt{7} +\sqrt{5}} \\\ \\ Similarly, \\\ \\ \sqrt{5} -\sqrt{3}= \frac{2}{\sqrt{5} +\sqrt{3}} \\\ \\ \sqrt{9} -\sqrt{7}= \frac{2}{\sqrt{9} +\sqrt{7}} \\\ \\ \sqrt{11} -\sqrt{9}= \frac{2}{\sqrt{11} +\sqrt{9}} \\\ \\ Among\ all\ of\ these\ \sqrt{5} -\sqrt{3} is\ greatest.\\\ \\

Question 03
The\ least\ one\ of\ 2\sqrt{3},\ 2\sqrt[4]{5},\ \sqrt{8} \ and\ 3\sqrt{2}\ is \\\ \\ (a) \sqrt{8}                (b) 2\sqrt[4]{5}\\\ \\ (c) 2\sqrt{3}                 (d) \ 3\sqrt{2} \\\ \\ Read Solution

2\sqrt{3} ,2\sqrt[4]{5} ,\sqrt{8} \ ,\ 3\sqrt{2} \\\ \\ ⟹ ( 4\times 3)^{\frac{1}{2}} ,( 5\times 16)^{\frac{1}{4}} ,( 8)^{\frac{1}{2}} ,( 2\times 9)^{\frac{1}{2}} \\\ \\ ⟹ ( 12)^{\frac{1}{2}} ,( 80)^{\frac{1}{4}} ,( 8)^{\frac{1}{2}} ,( 18)^{\frac{1}{2}} \\\ \\ ⟹ ( 12)^{\frac{2}{4}} ,( 80)^{\frac{1}{4}} ,( 8)^{\frac{2}{4}} ,( 18)^{\frac{2}{4}} \\\ \\ ⟹ \sqrt[4]{144} ,\sqrt[4]{80} ,\sqrt[4]{64} ,\sqrt[4]{324} \\\ \\ \sqrt{8} \ is\ smallest.\\\ \\

Option (a) is the right answer

Question 04
2\sqrt[3]{32} -3\sqrt[3]{4} +\sqrt[3]{500} =? \\\ \\ (a) 4\sqrt[3]{6}                      (b) 3\sqrt{24} \\\ \\ (c) 6\sqrt[5]{4}                      (d) 6\sqrt[3]{4} \\\ \\ Read Solution

2\sqrt[3]{32} -3\sqrt[3]{4} +\sqrt[3]{500} \\\ \\ ⟹ 2\sqrt[3]{2^{3} \times 4} -3\sqrt[3]{4} +\sqrt[3]{5^{3} \times 4} \\\ \\ ⟹ 4\sqrt[3]{4} -3\sqrt[3]{4} +5\sqrt[3]{4} \\\ \\ ⟹ 9\sqrt[3]{4} -3\sqrt[3]{4} \\\ \\ ⟹ 6\sqrt[3]{4}

Option (d) is the right answer

Question 05
\left(\sqrt{2} +\sqrt{7-2\sqrt{10}}\right)\ is\ equal\ to \\\ \\ (a) \sqrt{2}                (b) \sqrt{7}\\\ \\ (c) \sqrt{5}                 (d) 2\sqrt{5} \\\ \\ Read Solution

\left(\sqrt{2} +\sqrt{7-2\sqrt{10}}\right) \\\ \\ \Longrightarrow \ \sqrt{2} +\sqrt{\left(\sqrt{5}\right)^{2} +\left(\sqrt{2}\right)^{2} -2\sqrt{5}\sqrt{2}} \\\ \\ ⟹ \sqrt{2} +\sqrt{\left(\sqrt{5} -\sqrt{2}\right)^{2}} \\\ \\ ⟹ \sqrt{2} +\sqrt{5} -\sqrt{2} \\\ \\ ⟹ \sqrt{5} \\\ \\

Option (c) is the right answer

Question 06
If \ x+\frac{1}{x} =-2\ then\ value\ of\ x^{2n+1} +\frac{1}{x^{2n+1}}\ where\ n\ is\ a\ positive\ integer\ is \\\ \\ (a) 0                 (b) 2 \\\ \\ (c) -2\ \ \ \ \ \     (d) -5 \\\ \\ Read Solution

Let\ x=-1 \\\ \\ x+\frac{1}{x} =-2 \\\ \\ ⟹ -1+\frac{1}{-1} =-2 \ \ \ hence \ x=-1\\\ \\ Put \ n=\ 1 \\\ \\ x^{2n+1} +\frac{1}{x^{2n+1}} \\\ \\ ⟹ -1^{2\times 1+1} +\frac{1}{-1^{2\times 1+1}} \\\ \\ ⟹ -1^{3} +\frac{1}{-1^{3}} =-2 \\\ \\

Option (c) is the right answer

Question 07
( 256)^{0.16} \times ( 4)^{0.36} is\ equal\ to\\\ \\ (a) 64               (b) 16 \\\ \\ (c) 256.25        (d) 4 \\\ \\ Read Solution

⟹ ( 256)^{0.16} \times ( 4)^{0.36} \\\ \\ ⟹ 4^{4\times 0.16} \times 4^{0.36} \\\ \\ ⟹ 4^{0.64} \times 4^{0.36} \\\ \\ ⟹ 4^{1}=4 \\\ \\

Option (d) is the right answer

Question 08
\left(\frac{2}{\sqrt{6} +2} +\frac{1}{\sqrt{7} +\sqrt{6}} +\frac{1}{\sqrt{8} -\sqrt{7}} +2-2\sqrt{2}\right) is\ equal\ to \\\ \\ (a) 2\sqrt{7}               (b) 2\sqrt{2} \\\ \\(c) \sqrt{2}\ \ \ \ \ \ \ \ \      (d) 0 \\\ \\ Read Solution

\left(\frac{2}{\sqrt{6} +2} +\frac{1}{\sqrt{7} +\sqrt{6}} +\frac{1}{\sqrt{8} -\sqrt{7}} +2-2\sqrt{2}\right) \\\ \\ ⟹ \left(\frac{2}{\sqrt{6} +2} \times \frac{\sqrt{6} -2}{\sqrt{6} -2} +\frac{1}{\sqrt{7} +\sqrt{6}} \times \frac{\sqrt{7} -\sqrt{6}}{\sqrt{7} -\sqrt{6}} +\frac{1}{\sqrt{8} -\sqrt{7}} \times \frac{\sqrt{8} +\sqrt{7}}{\sqrt{8} +\sqrt{7}} +2-2\sqrt{2}\right) \\\ \\ ⟹ \left(\frac{2\left(\sqrt{6} -2\right)}{\left(\sqrt{6} +2\right)\left(\sqrt{6} -2\right)} +\frac{1\left(\sqrt{7} -\sqrt{6}\right)}{\left(\sqrt{7} +\sqrt{6}\right)\left(\sqrt{7} -\sqrt{6}\right)} +\frac{1\left(\sqrt{8} +\sqrt{7}\right)}{\left(\sqrt{8} -\sqrt{7}\right)\left(\sqrt{8} +\sqrt{7}\right)} +2-2\sqrt{2}\right) \\\ \\ ⟹ \left(\frac{2\left(\sqrt{6} -2\right)}{6-4} +\frac{\left(\sqrt{7} -\sqrt{6}\right)}{7-6} +\frac{\left(\sqrt{8} +\sqrt{7}\right)}{8-7} +2-2\sqrt{2}\right) \\\ \\ ⟹ \left(\frac{2\left(\sqrt{6} -2\right)}{2} +\frac{\left(\sqrt{7} -\sqrt{6}\right)}{1} +\frac{\left(\sqrt{8} +\sqrt{7}\right)}{1} +2-2\sqrt{2}\right) \\\ \\ ⟹ \sqrt{6} -2+\sqrt{7} -\sqrt{6} +\sqrt{8} +\sqrt{7} +2-2\sqrt{2} \\\ \\ ⟹\sqrt{6} -2+\sqrt{7} -\sqrt{6} +2\sqrt{2} +\sqrt{7} +2-2\sqrt{2} \\\ \\ ⟹ 2\sqrt{7} \\\ \\

option (d) is the right answer

Question 09
The\ value\ of\ \frac{2+\sqrt{3}}{2-\sqrt{3}} +\frac{2-\sqrt{3}}{2+\sqrt{3}} +\frac{\sqrt{3} +1}{\sqrt{3} -1} is \\\ \\ (a) 16+\sqrt{3}               (b) 4+\sqrt{3} \\\ \\ (c) 2-\sqrt{3}             (d) 2+\sqrt{3} \\\ \\ Read Solution

\frac{2+\sqrt{3}}{2-\sqrt{3}} +\frac{2-\sqrt{3}}{2+\sqrt{3}} +\frac{\sqrt{3} +1}{\sqrt{3} -1} \\\ \\ ⟹\left(\frac{2+\sqrt{3}}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}}\right) +\left(\frac{2-\sqrt{3}}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}\right) +\left(\frac{\sqrt{3} +1}{\sqrt{3} -1} \times \frac{\sqrt{3} +1}{\sqrt{3} +1}\right) \\\ \\ ⟹ \left(\frac{\left( 2+\sqrt{3}\right)^{2}}{\left( 2-\sqrt{3}\right)\left( 2+\sqrt{3}\right)}\right) +\left(\frac{\left( 2-\sqrt{3}\right)^{2}}{\left( 2+\sqrt{3}\right)\left( 2-\sqrt{3}\right)}\right) +\left(\frac{\left(\sqrt{3} +1\right)^{2}}{\left(\sqrt{3} +1\right)\left(\sqrt{3} -1\right)}\right) \\\ \\ ⟹ \left(\frac{4+3+4\sqrt{3}}{4-3}\right) +\left(\frac{4+3-4\sqrt{3}}{4-3}\right) +\left(\frac{3+1+2\sqrt{3}}{3-1}\right) \\\ \\ ⟹ \ 4+3+4\sqrt{3} +4+3-4\sqrt{3} +\left(\frac{4+2\sqrt{3}}{2}\right) \\\ \\ ⟹ \ 7+7+2\left(\frac{2+\sqrt{3}}{2}\right) \\\ \\ ⟹ 7+7+2+\sqrt{3} \\\ \\ ⟹ 16+\sqrt{3} \\\ \\

Option (a) is the right answer

Question 10
The\ value\ of\ \frac{1}{\sqrt{2} +1} +\frac{1}{\sqrt{3} +\sqrt{2}} +\frac{1}{\sqrt{4} +\sqrt{3}} +....+\frac{1}{\sqrt{100} +\sqrt{99}} is \\\ \\ (a) 1                             (b) 9 \\\ \\ (c) \sqrt{199}\ \              (d) \sqrt{99} -1\\\ \\ Read Solution

⟹ \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{3} +\sqrt{2}}= \sqrt{3} -\sqrt{2} \\\ \\ And\ so\ on. \\\ \\ \\\ \\ Now\ put\ the\ values \\\ \\ ⟹ \sqrt{2} -1+\sqrt{3} -\sqrt{2} \ +\sqrt{4} -\sqrt{3} +....+\sqrt{100} -\sqrt{99} \\\ \\ ⟹ \sqrt{100} -1 \\\ \\ ⟹ 10-1=9 \\\ \\

Option (b) is the right answer