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Home > Practice > Arithmetic Aptitude > Surds and Indices > Miscellaneous

31. Simplify : (3)⁸ × (3)⁴ = ?

A. (27)³

B. (27)⁵

C. (729)²

D. (729)³

Discuss

Solution:

3⁸ × 3⁴
= 3^{(8 + 4)}
= 3¹²
= (3⁶)²
= (729)²

32. Simplify : $\frac{343 \times 49}{216 \times 16 \times 81} = ?$

A. $\frac{7^{5}}{6^{7}}$

B. $\frac{7^{5}}{6^{8}}$

C. $\frac{7^{6}}{6^{7}}$

D. $\frac{7^{4}}{6^{8}}$

Discuss

Solution:

\begin{aligned} \frac{343 \times 49}{216 \times 16 \times 81} \\ = \frac{7^{3} \times 7^{2}}{6^{3} \times 2^{4} \times 3^{4}} \\ = \frac{7^{(3 + 2)}}{6^{3} \times {(2 \times 3)}^{4}} \\ = \frac{7^{5}}{6^{3} \times 6^{4}} \\ = \frac{7^{5}}{6^{(3 + 4)}} \\ = \frac{7^{5}}{6^{7}} \end{aligned}

33. Simplify : $\frac{16 \times 32}{9 \times 27 \times 81} = ?$

A. ${(\frac{2}{3})}^{9}$

B. ${(\frac{2}{3})}^{11}$

C. ${(\frac{2}{3})}^{12}$

D. ${(\frac{2}{3})}^{13}$

Discuss

Solution:

\begin{aligned} \frac{16 \times 32}{9 \times 27 \times 81} \\ = \frac{2^{4} \times 2^{5}}{3^{2} \times 3^{3} \times 3^{4}} \\ = \frac{2^{(4 + 5)}}{3^{(2 + 3 + 4)}} \\ = \frac{2^{9}}{3^{9}} \\ = {(\frac{2}{3})}^{9} \end{aligned}

34. Given $\sqrt{2}$ = 1.414, the value of $\sqrt{8}$ $+$ $2 \sqrt{32}$ $-$ $3 \sqrt{128}$ $+$ $4 \sqrt{50}$ is = ?

A. 8.484

B. 8.526

C. 8.426

D. 8.876

Discuss

Solution:

\begin{aligned} \sqrt{2} = 1.414 \\ \Rightarrow \sqrt{8} + 2 \sqrt{32} - 3 \sqrt{128} + 4 \sqrt{50} \\ \Rightarrow 2 \sqrt{2} + 2 \times 4 \sqrt{2} - 3 \times 8 \sqrt{2} + 4 \times 5 \sqrt{2} \\ \Rightarrow 2 \sqrt{2} + 8 \sqrt{2} - 24 \sqrt{2} + 20 \sqrt{2} \\ \Rightarrow 6 \sqrt{2} \\ \Rightarrow 6 \times 1.414 \\ \Rightarrow 8.484 \end{aligned}

35. $9^{3} \times {(81)}^{2} \div {(27)}^{3} = {(3)}^{?}$

A. 3

B. 4

C. 5

D. 6

Discuss

Solution:

\begin{aligned} Let 9^{3} \times {(81)}^{2} \div {(27)}^{3} = {(3)}^{x} then \\ \Rightarrow {(3)}^{x} = \frac{{(3^{2})}^{3} \times {(3^{4})}^{2}}{{(3^{3})}^{3}} \\ \Rightarrow {(3)}^{x} = \frac{3^{(2 \times 3)} \times 3^{(4 \times 2)}}{3^{(3 \times 3)}} \\ \Rightarrow {(3)}^{x} = \frac{3^{6} \times 3^{8}}{3^{9}} \\ \Rightarrow {(3)}^{x} = \frac{3^{(6 + 8)}}{3^{9}} \\ \Rightarrow {(3)}^{x} = \frac{3^{14}}{3^{9}} \\ \Rightarrow {(3)}^{x} = 3^{(14 - 9)} \\ \Rightarrow {(3)}^{x} = 3^{5} \\ \Rightarrow {(3)}^{x} = 5 \end{aligned}

36. ${(6)}^{4} \div {(36)}^{3} \times 216 = 6^{(? - 5)}$

A. 1

B. 4

C. 6

D. 7

Discuss

Solution:

\begin{aligned} Let , \\ {(6)}^{4} \div {(36)}^{3} \times 216 = 6^{(x - 5)} \\ Then, \\ 6^{(x - 5)} = {(6)}^{4} \div {(6^{2})}^{3} \times 6^{3} \\ \Rightarrow 6^{(x - 5)} = 6^{4} \div 6^{(2 \times 3)} \times 6^{3} \\ \Rightarrow 6^{(x - 5)} = 6^{4} \div 6^{6} \times 6^{3} \\ \Rightarrow 6^{(x - 5)} = 6^{(4 - 6 + 3)} \\ \Rightarrow 6^{(x - 5)} = 6 \\ \Rightarrow x - 5 = 1 \\ \Rightarrow x = 6 \end{aligned}

37. The value of ${(256)}^{\frac{5}{4}}$ is = ?

A. 512

B. 984

C. 1024

D. 1032

Discuss

Solution:

\begin{aligned} {(256)}^{\frac{5}{4}} \\ = {(4^{4})}^{\frac{5}{4}} \\ = 4^{(4 \times \frac{5}{4})} \\ = 4^{5} \\ = 1024 \end{aligned}

38. $\sqrt{2 + \sqrt{2 + \sqrt{2 + . . . . . .}}}$ is equal to ?

A. $\sqrt{2}$

B. $2 \sqrt{2}$

C. 2

D. 3

Discuss

Solution:

\begin{aligned} x = \sqrt{2 + \sqrt{2 + \sqrt{2 + . . . . . .}}} \\ \Rightarrow x^{2} = 2 + \sqrt{2 + \sqrt{2 + . . . . . . .}} \\ \Rightarrow x^{2} = 2 + x \\ \Rightarrow x^{2} - x - 2 = 0 \\ \Rightarrow x (x - 2) + 1 (x - 2) = 0 \\ \Rightarrow (x + 1) (x - 2) = 0 \\ \Rightarrow x = 2 \end{aligned}

39. The value of $2 + \sqrt{0.09} - \sqrt[3]{0.008} - 75 %$ of 2.80 is = ?

A. 0

B. 0.01

C. -1

D. 0.001

Discuss

Solution:

\begin{aligned} 2 + \sqrt{0.09} - \sqrt[3]{0.008} - 75 % of 2.80 \\ = 2 + 0.3 - 0.2 - (\frac{3}{4} \times 2.8) \\ = 2 + 0.3 - 0.2 - 2.10 \\ = 2.3 - 2.3 \\ = 0 \end{aligned}

40. The value of ${(3 + 2 \sqrt{2})}^{- 3}$ $+ {(3 - 2 \sqrt{2})}^{- 3} = ?$

A. 189

B. 180

C. 108

D. 198

Discuss

Solution:

\begin{aligned} {(3 + 2 \sqrt{2})}^{- 3} + {(3 - 2 \sqrt{2})}^{- 3} \\ = {(\frac{1}{3 + 2 \sqrt{2}})}^{3} + {(\frac{1}{3 - 2 \sqrt{2}})}^{3} \end{aligned}

= {(\frac{1}{(3 - 2 \sqrt{2})} \times \frac{3 + 2 \sqrt{2}}{3 + 2 \sqrt{2}})}^{3} +

{(\frac{1}{(3 + 2 \sqrt{2})} \times \frac{3 - 2 \sqrt{2}}{3 - 2 \sqrt{2}})}^{3}

\begin{aligned} = {(\frac{3 - 2 \sqrt{2}}{9 - 8})}^{3} + {(\frac{3 + 2 \sqrt{2}}{9 - 8})}^{3} \\ = {(3 - 2 \sqrt{2})}^{3} + {(3 + 2 \sqrt{2})}^{3} \\ a = 3 - 2 \sqrt{2} \\ b = 3 + 2 \sqrt{2} \\ [∵ a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - a b)] \\ = (3 - 2 \sqrt{2} + 3 + 2 \sqrt{2}) (17 + 17 - 1) \\ = (6) (33) \\ = 198 \end{aligned}

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