# Power, Indices and Surds – Aptitude Question

Question (01)
How\ much\ does\ \sqrt{12} +\sqrt{18} \ exceeds\ \sqrt{3} +\sqrt{2} \ \ ?\\\ \\ ( a) \ 2\left(\sqrt{3} -\sqrt{2}\right)\\ \\ ( b) \ 2\left(\sqrt{3} +\sqrt{2}\right)\\ \\ ( c) \ \sqrt{3} +2\sqrt{2})\\ \\ ( d) \ \sqrt{3} -2\sqrt{2})\\\ \\ Read Solution

Solution

( \sqrt{12} +\sqrt{18})-( \sqrt{3} +\sqrt{2})\\\ \\ ⟹ 2\sqrt{3}+3 \sqrt{2}- \sqrt{3} - \sqrt{2} \\\ \\ ⟹ \sqrt{3} +2\sqrt{2}

Option (b) is the right answer

Question (02)
Find\ the\ value\ of\ \sqrt{5+2\sqrt{6}} -\frac{1}{\sqrt{5+2\sqrt{6}}} ?\\\ \\ ( a)2\sqrt{3}\\ \\ ( b) 2\sqrt{2}\\ \\ ( c) \ 1+\sqrt{5}\\ \\ ( d) \ \sqrt{5} -1 \\\ \\ Read Solution

a^{2} +b^{2} +2ab=( a+b)^{2} \\\ \\ \sqrt{5+2\sqrt{6}} =\sqrt{\left(\sqrt{3} +\sqrt{2}\right)}^{2}= \sqrt{3} +\sqrt{2} \\\ \\ ⟹ \sqrt{5+2\sqrt{6}} - \frac{1}{\sqrt{5+2\sqrt{6}}} \\\ \\ ⟹ \sqrt{3} +\sqrt{2} - \frac{1}{\sqrt{3} +\sqrt{2}} \\\ \\ ⟹ \sqrt{3} +\sqrt{2}-( \frac{1}{\sqrt{3} +\sqrt{2}} \times \frac{\sqrt{3} -\sqrt{2}}{\sqrt{3} -\sqrt{2}}) \\\ \\ ⟹ \sqrt{3} +\sqrt{2}-( \frac{\sqrt{3} -\sqrt{2}}{3-2}) \\\ \\ ⟹ \sqrt{3} +\sqrt{2}- \sqrt{3} +\sqrt{2} \\\ \\⟹ 2 \sqrt{2}

Option (b) is the right answer

Question (03)
Find\ the\ value\ of\ \sqrt{2^{4}} +\sqrt[3]{64} +\sqrt[4]{2^{8}}\\ \\ ( a) 12 \\ \\ ( b) 16 \\ \\ ( c) 18 \\ \\ ( d) 24 \\\ \\ Read Solution

\sqrt{2^{4}} +\sqrt[3]{64} +\sqrt[4]{2^{8}} \\\ \\ ⟹ 2^{4\times \frac{1}{2}} +4^{3\times \frac{1}{3}} +2^{8\times \frac{1}{4}} \\\ \\ ⟹ 2^{2} +4+2^{2} \\\ \\⟹ 4+4+4=12

Option (a) is the right answer

Question (04)
Find\ the\ value\ of\ 2\sqrt[3]{32} +3\sqrt[3]{4} +\sqrt[3]{500} \\ \\ ( a) 4\sqrt[3]{6}\\ \\ ( b) 3\sqrt{24} \\ \\ ( c) 6\sqrt[3]{4} \\ \\ ( d) 916 \\\ \\ Read Solution

2\sqrt[3]{32} -3\sqrt[3]{4} +\sqrt[3]{500} \\\ \\ ⟹ 2 \sqrt[3]{2\times 2\times 2\times 4} -3\sqrt[3]{4} +\sqrt[3]{5\times 5\times 5\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 (c) is the right answer

(05) Find the solution of
\frac{\frac{3}{2+\sqrt{3}} -\frac{2}{2-\sqrt{3}}}{2-5\sqrt{3}} \\\ \\

(a) 1
(b) 2
(c) 3
(d) 4

Read Solution

\frac{\frac{3}{2+\sqrt{3}} -\frac{2}{2-\sqrt{3}}}{2-5\sqrt{3}} \\\ \\ ⟹ \frac{\frac{3\left( 2-\sqrt{3}\right) -2\left( 2+\sqrt{3}\right)}{\left( 2+\sqrt{3}\right)\left( 2-\sqrt{3}\right)}}{2-5\sqrt{3}} \\\ \\⟹ \frac{6-3\sqrt{3} -4-2\sqrt{3}}{\left( 2+\sqrt{3}\right)\left( 2-\sqrt{3}\right)\left( 2-5\sqrt{3}\right)} \\\ \\⟹ \frac{2-5\sqrt{3}}{2-5\sqrt{3}}=1

Option (a) is the right answer

Question (06)
Which\ of\ the\ following\ is\ least\\ \\ \sqrt{3} ,\ \sqrt[3]{2} ,\ \sqrt{2} \ and\ \sqrt[3]{4}\\\ \\ ( a) \sqrt{3} \\ \\ ( b) \sqrt[3]{2} ,\\ \\ ( c) \sqrt{2} \\ \\ ( d) \sqrt[3]{4}\\\ \\ Read Solution

\sqrt{3} ,\sqrt[3]{2} ,\sqrt{2} ,\sqrt[3]{4} \\\ \\⟹ 3^{\frac{1}{2}} ,2^{\frac{1}{3}} ,2^{\frac{1}{2}}, 4^{\frac{1}{3}} \\\ \\⟹ 3^{\frac{3}{6}} ,2^{\frac{2}{6}} ,2^{\frac{3}{6}}, 4^{\frac{2}{6}} \ \ \ \             (LCM\ of\ 3\ and\ 6\ is\ 6) \\\ \\⟹ \sqrt[6]{3^{3}} ,\sqrt[6]{2^{2}} ,\sqrt[6]{2^{3}} ,\sqrt[6]{4^{2}} \\\ \\⟹ \sqrt[6]{27} ,\sqrt[6]{4} ,\sqrt[6]{8} ,\sqrt[6]{16} \\\ \\ Hence \sqrt[3]{2} \ is\ least.

option (b) is the right answer

Question (07)
Which\ of\ the\ following\ is\ biggest\\ \\ \sqrt[3]{4} ,\ \sqrt[4]{6} ,\ \sqrt[6]{15} \ and\ \sqrt[12]{24} 5\\\ \\ (a) \sqrt[3]{4} \\ \\ (b) \sqrt[4]{6} \\ \\ (c) \sqrt[6]{15} \\ \\ (d) \sqrt[12]{245}\\\ \\

Read Solution

\sqrt[3]{4} ,\sqrt[4]{6} ,\sqrt[6]{15} ,\sqrt[12]{245} \\\ \\⟹ 4^{\frac{1}{3}} ,6^{\frac{1}{4}} ,15^{\frac{1}{6}} ,245^{\frac{1}{12}}\\\ \\⟹ 4^{\frac{4}{12}} ,6^{\frac{3}{12}} ,15^{\frac{2}{12}}, 245^{\frac{1}{12}} \ \ \ \                   (LCM of 3,4,6 and12 is 12) \\\ \\ ⟹ \sqrt[12]{4^{4}} ,\sqrt[12]{6^{3}} ,\sqrt[12]{15^{2}} ,\sqrt[12]{245}\\\ \\⟹ \sqrt[12]{256} ,\sqrt[12]{216} ,\sqrt[12]{225} ,\sqrt[12]{245} \\\ \\ Hence \sqrt[3]{4} \ is \ biggest.

option (a) is the right answer

Question (08)
\sqrt{8} \ -\sqrt{4} \ -\sqrt{2} \ equals?\\\ \\ Read Solution

( \sqrt{8} ,\sqrt{4} -\sqrt{2}) \\\ \\ ⟹ 2\sqrt{2} -2-\sqrt{2} \\\ \\ ⟹ 2\sqrt{2} -\sqrt{2} -2 \\\ \\ ⟹ \sqrt{2} -2

Question (09)
{64^{\frac{-2}{3}} \times \ \frac{1}{4}^{-2}} \ equals?\\\ \\ Read Solution

64^{\frac{-2}{3}} \times \left(\frac{1}{4}\right)^{-2} \\\ \\ ⟹ \left( 4^{3}\right)^{\frac{-2}{3}} \times \left(\frac{1}{4}\right)^{-2} \\\ \\ ⟹ 4^{-2} \times \left(\frac{1}{4}\right)^{-2} \\\ \\ ⟹ \left(\frac{1}{4}\right)^{2}\times \left(\frac{1}{4}\right)^{-2} \\\ \\ ⟹ \left(\frac{1}{4}\right)^{2-2} \\\ \\⟹ \left(\frac{1}{4}\right)^{0}=1 \\\ \\

Question 10
\frac{2+\sqrt{3}}{2-\sqrt{3}} \ \ +\ \frac{2-\sqrt{3}}{2+\sqrt{3}} \ +\frac{\sqrt{3} -1}{\sqrt{3} +1}\\\ \\ (a) 2-\sqrt{3}\\ \\ (b) 2+\sqrt{3}\\ \\ (c) 16-\sqrt{3}\\ \\ (d) 40-\sqrt{3}\\ \\ Read Solution

\left(\frac{2+\sqrt{3}}{2-\sqrt{3}} +\frac{2-\sqrt{3}}{2+\sqrt{3}} +\frac{\sqrt{3} -1}{\sqrt{3} +1}\right)\\\ \\ ⟹ \left(\frac{\left( 2+\sqrt{3}\right)^{2} +\left( 2-\sqrt{3}\right)^{2}}{\left( 2+\sqrt{3}\right)\left( 2-\sqrt{3}\right)} +\frac{\sqrt{3} -1}{\sqrt{3} +1} \times \frac{\sqrt{3} -1}{\sqrt{3} -1}\right) \\\ \\ ⟹ \left(\frac{4+3+4\sqrt{3} +4+3-4\sqrt{3}}{4-2\sqrt{3} +2\sqrt{3} -3} +\frac{\left(\sqrt{3} -1\right)^{2}}{3-1}\right) \\\ \\ ⟹ \left( 14+\frac{3+1-2\sqrt{3}}{2}\right) \\\ \\ ⟹ \left( 14+\frac{4-2\sqrt{3}}{2}\right) \\\ \\ ⟹ 14+ \frac{2\left( 2-\sqrt{3}\right)}{2} \\\ \\ ⟹ 14+ 2-\sqrt{3} \\\ \\ ⟹ 16- \sqrt{3} \\\ \\

option (c) is the right answer

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