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# Power, Indices and Surds MCQ Question with Answer

## Power, Indices and Surds: Quantitative Aptitude MCQ Question with Answer

Q1. Evaluate $\frac{\sqrt{24}+\sqrt{6}}{\sqrt{24}-\sqrt{6}}$

(a) 5

(b) 4

(c) 3

(d) 2

 Answer: (c)Solution: $\frac{\sqrt{24}+\sqrt{6}}{\sqrt{24}-\sqrt{6}}=\frac{2\sqrt{6}+\sqrt{6}}{2\sqrt{6}-\sqrt{6}}=\frac{3\sqrt{6}}{\sqrt{6}}=3$

Q2. The smallest of is: $\sqrt{8}+\sqrt{5},\sqrt{7}+\sqrt{6},\sqrt{10}+\sqrt{3}\mathrm{and} \sqrt{11}+\sqrt{2}$

(a) $\sqrt{8}+\sqrt{5}$

(b) $\sqrt{7}+\sqrt{6}$

(c) $\sqrt{10}+\sqrt{3}$

(d) $\sqrt{11}+\sqrt{2}$

 Answer: (d)Solution: $\sqrt{\left( \sqrt{8}+\sqrt{5} \right) ^2}=\sqrt{13+2\sqrt{40}}$$\sqrt{\left( \sqrt{7}+\sqrt{6} \right) ^2}=\sqrt{13+2\sqrt{42}}$$\sqrt{\left( \sqrt{10}+\sqrt{3} \right) ^2}=\sqrt{13+2\sqrt{30}}$$\sqrt{\left( \sqrt{11}+\sqrt{2} \right) ^2}=\sqrt{13+2\sqrt{22}}Since 2\sqrt{22} is smallest then \sqrt{11}+\sqrt{2} is smallest Q3. Find the value of \sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}} (a) 2 (b) 2^{\frac{1}{2}} (c) 2^{\frac{11}{12}} (d) 2^{\frac{31}{32}}  Answer: (d)Solution: \sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}=2^{\frac{2^5-1}{2^5}}=2^{\frac{31}{32}}Rules: \sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}}}}}}=\mathrm{X}^{\frac{\mathrm{X}^{\mathrm{n}}-1}{\mathrm{X}^{\mathrm{n}}}}where n is no. of times X repeated. Q4. Find the value of \sqrt{5\sqrt{5\sqrt{5…\mathrm{\alpha}}}} (a) 5 (b) 5^{\frac{7}{8}} (c) 5^{\frac{1}{8}} (d) 5^{\frac{1}{3}}  Answer: (a)Solution: \sqrt{5\sqrt{5\sqrt{5…\mathrm{\alpha}}}}=5Rules: \sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}\sqrt{\mathrm{X}}}}}}=\mathrm{X} Q5. Find the value of \sqrt{12+\sqrt{12+\sqrt{12….}}} (a) 3 (b) –3 (c) 2 (d) –2  Answer: (b)Solution:\sqrt{12+\sqrt{12+\sqrt{12….}}}=\mathrm{x}\left( \mathrm{let} \right) \Rightarrow 12 +\mathrm{x} =\,\,\mathrm{x}^2 ( square both side) \Rightarrow \mathrm{x}^2-\mathrm{x}-12=0 \Rightarrow \mathrm{x}^2-4\mathrm{x}+3\mathrm{x}-12=0 \Rightarrow \mathrm{x}\left( \mathrm{x}-4 \right) +3\left( \mathrm{x}-12 \right) =0 \Rightarrow \left( \mathrm{x}-4 \right) \left( \mathrm{x}+3 \right) =0∴ x = 4 or -3Here –3 is in option.Rules: \sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) +\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) +\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) ….}}}=-\mathrm{n},\left( \mathrm{n}+1 \right) Q6. Find the value of \sqrt{30-\sqrt{30-\sqrt{30….}}} (a) 5 (b) 6 (c) –5 (d) 4  Answer: (a)Solution: \sqrt{30-\sqrt{30-\sqrt{30….}}}=\mathrm{x}\left( \mathrm{let} \right) \Rightarrow 30 -\mathrm{x} =\,\,\mathrm{x}^2 ( square both side) \Rightarrow \mathrm{x}^2+\mathrm{x}-30=0 \Rightarrow \mathrm{x}^2+6\mathrm{x}-5\mathrm{x}-30=0 \Rightarrow \mathrm{x}\left( \mathrm{x}+6 \right) -5\left( \mathrm{x}+6 \right) =0 \Rightarrow \left( \mathrm{x}+6 \right) \left( \mathrm{x}-5 \right) =0∴ x = -6 or 5Here 5 is in option.Rules: \sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) -\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) -\sqrt{\mathrm{n}\left( \mathrm{n}+1 \right) ….}}}=\mathrm{n},-\left( \mathrm{n}+1 \right) Q7. Find the square root of 105\frac{4}{64} (a) 15\frac{1}{4} (b) 15\frac{3}{4} (c) 10\frac{1}{4} (d) 6\frac{1}{4}  Answer: (c)Solution: \sqrt{105\frac{4}{64}\,\,}=\,\,\sqrt{\frac{6724}{64}}=\frac{82}{8}=10\frac{1}{4} Q8. Find the value of \frac{1}{\sqrt{9}-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-\sqrt{4}} (a) 0 (b) 1 (c) 1/3 (d) 5 (e) 1/5  Answer: (d)Solution: \frac{1}{\sqrt{9}-\sqrt{8}}=\frac{\sqrt{9}+\sqrt{8}}{\left( \sqrt{9}-\sqrt{8} \right) \left( \sqrt{9}+\sqrt{8} \right)}=\sqrt{9}+\sqrt{8}Similarly \frac{1}{\sqrt{8}-\sqrt{7}}=\sqrt{8}+\sqrt{7}$$\frac{1}{\sqrt{7}-\sqrt{6}}=\sqrt{7}+\sqrt{6}$$\frac{1}{\sqrt{6}-\sqrt{5}}=\sqrt{6}+\sqrt{5}$$\frac{1}{\sqrt{5}-\sqrt{4}}=\sqrt{5}+\sqrt{4}$$\therefore \sqrt{9}+\sqrt{8}-\sqrt{8}-\sqrt{7}+\sqrt{7}-\sqrt{6}-\sqrt{5}+\sqrt{5}+\sqrt{4}$= 3 + 2 = 5

Q9. The value of $\left( \frac{1024}{243} \right) ^{-\frac{4}{5}}$ is:

(a) $\frac{81}{16}$

(b) $\frac{81}{256}$

(c) $\frac{4}{9}$

(d) $\frac{9}{4}$

(e) None of these

 Answer: (b)Solution:$\left( \frac{1024}{243} \right) ^{-\frac{4}{5}}\,\,=\,\,\left( \frac{\left( 2 \right) ^{10}}{\left( 3 \right) ^5} \right) ^{^{-\frac{4}{5}}}=\,\,\frac{2^{10\times \left( -\frac{4}{5} \right)}}{3^{5\times \left( -\frac{4}{5} \right)}}\,\,=\,\,\frac{2^{-8}}{3^{-4}}=\left( \frac{3}{4} \right) ^4=\frac{81}{256}$

Q10. The value of $\left( \sqrt{125} \right) ^{\frac{1}{3}}$is:

(a) 2

(b) 4

(c) 5

(d) 8

(e) None of these

 Answer: (e)Solution: $\left( \sqrt{125} \right) ^{\frac{1}{3}}\,\,=\,\,\left( \sqrt{5^3} \right) ^{\frac{1}{3}}\,\,=\,\,\left( 5^{\frac{3}{2}} \right) ^{\frac{1}{3}}=\sqrt{5}$

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