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71. $[\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}]$ simplifies to = ?

A. $2 \sqrt{6}$

B. $4 \sqrt{6}$

C. $2 \sqrt{3}$

D. $3 \sqrt{2}$

Discuss

Solution:

[\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}]

= \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \times

\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} -

\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \times

\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}

\begin{aligned} = \frac{{(\sqrt{3} + \sqrt{2})}^{2}}{3 - 2} - \frac{{(\sqrt{3} - \sqrt{2})}^{2}}{3 - 2} \\ = (3 + 2 + 2 \sqrt{6}) - (3 + 2 - 2 \sqrt{6}) \\ = 4 \sqrt{6} \end{aligned}

72. If $\sqrt{3}$ = 1.732 is given, then the value of $\frac{2 + \sqrt{3}}{2 - \sqrt{3}}$ is = ?

A. 11.732

B. 13.928

C. 12.928

D. 13.925

Discuss

Solution:

\begin{aligned} \sqrt{3} = 1.732 \\ \Rightarrow \frac{2 + \sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} \\ \Rightarrow \frac{{(2 + \sqrt{3})}^{2}}{4 - 3} \\ \Rightarrow 4 + 3 + 4 \sqrt{3} \\ \Rightarrow 7 + 4 \times 1.732 \\ \Rightarrow 7 + 6.928 \\ \Rightarrow 13.928 \end{aligned}

73. Evaluate: $16 \sqrt{\frac{3}{4}} - 9 \sqrt{\frac{4}{3}}$ if $\sqrt{12}$ = 3.46

A. 3.46

B. 10.38

C. 13.84

D. 24.22

Discuss

Solution:

\begin{aligned} 16 \sqrt{\frac{3}{4}} - 9 \sqrt{\frac{4}{3}} \\ \Rightarrow 16 \sqrt{\frac{3 \times 4}{4 \times 4}} - 9 \sqrt{\frac{3 \times 4}{3 \times 3}} \\ \Rightarrow 16 \times \frac{\sqrt{12}}{4} - \frac{9 \sqrt{12}}{3} \\ \Rightarrow 4 \sqrt{12} - 3 \sqrt{12} \\ \Rightarrow \sqrt{12} \\ \Rightarrow 3.46 \end{aligned}

74. $2^{3.6} \times 4^{3.6} \times 4^{3.6} \times (32)^{2.3} =$ ${(32)}^{?}$

A. 5.9

B. 7.7

C. 9.5

D. 13.1

Discuss

Solution:

\begin{aligned} Let 2^{3.6} \times 4^{3.6} \times 4^{3.6} \times (32)^{2.3} = {(32)}^{x} \\ Then, 2^{3.6} \times {(2^{2})}^{3.6} \times {(2^{2})}^{3.6} \times (2^{5})^{2.3} = {(2^{5})}^{x} \\ \Leftrightarrow 2^{3.6} \times 2^{(2 \times 3.6)} \times 2^{(2 \times 3.6)} \times (2^{5})^{2.3} = {(2^{5})}^{x} \\ \Leftrightarrow 2^{(3.6 + 7.2 + 7.2)} \times (2^{5})^{2.3} = {(2^{5})}^{x} \\ \Leftrightarrow 2^{18} \times (2^{5})^{2.3} = {(2^{5})}^{x} \\ \Leftrightarrow {(2^{5})}^{3.6} \times (2^{5})^{2.3} = {(2^{5})}^{x} \\ \Leftrightarrow {(2^{5})}^{(3.6 + 2.3)} = {(2^{5})}^{x} \\ \Leftrightarrow {(2^{5})}^{5.9} = {(2^{5})}^{x} \\ \Leftrightarrow x = 5.9 \end{aligned}

75. $25^{2.7} \times 5^{4.2} \div 5^{5.4} = 25^{?}$

A. 1.6

B. 1.7

C. 3.2

D. 3.6

Discuss

Solution:

\begin{aligned} Let 25^{2.7} \times 5^{4.2} \div 5^{5.4} = 25^{x} \\ Then, 25^{2.7} \times 5^{(4.2 - 5.4)} = 25^{x} \\ \Leftrightarrow 25^{2.7} \times 5^{(- 1.2)} = 25^{x} \\ \Leftrightarrow 25^{2.7} \times \frac{1}{5^{1.2}} = 25^{x} \\ \Leftrightarrow \frac{25^{2.7}}{{(5^{2})}^{0.6}} = 25^{x} \\ \Leftrightarrow \frac{{(25)}^{2.7}}{{(25)}^{0.6}} = 25^{x} \\ \Leftrightarrow 25^{x} = 25^{(2.7 - 0.6)} = 25^{2.1} \\ \Leftrightarrow x = 2.1 \end{aligned}

76. $8^{2.4} \times 2^{3.7} \div {(16)}^{1.3} = 2^{?}$

A. 4.8

B. 5.7

C. 5.8

D. 7.1

Discuss

Solution:

\begin{aligned} Let 8^{2.4} \times 2^{3.7} \div {(16)}^{1.3} = 2^{x} \\ Then, {(2^{3})}^{2.4} \times 2^{3.7} \div {(2^{4})}^{1.3} = 2^{x} \\ \Leftrightarrow 2^{(3 \times 2.4)} \times 2^{3.7} \div 2^{(4 \times 1.3)} = 2^{x} \\ \Leftrightarrow 2^{7.2} \times 2^{3.7} \div 2^{5.2} = 2^{x} \\ \Leftrightarrow 2^{x} = 2^{(7.2 + 3.7 - 5.2)} \\ \Leftrightarrow 2^{x} = 2^{5.7} \\ \Leftrightarrow x = 5.7 \end{aligned}

77. If 3^(x+y) = 81 and 81^(x-y) = 3, then the value of x is = ?

A. 42

B. $\frac{15}{8}$

C. $\frac{17}{8}$

D. 39

Discuss

Solution:

\begin{aligned} 3^{x + y} =  81 \\ 3^{x + y} = 3^{4} \\ x + y = 4..... (i) \\ 8 1^{x - y} = 3 \\ 3^{4 x - 4 y} = 3^{1} \\ 4 x - 4 y =  1 . . . . (ii) \\ From equation (i) and (ii) \\ 4 x - 4 y = 1 \\ 4 x + 4 y = 16 \\ 8 x = 17 \\ x = \frac{17}{8} \end{aligned}

78. Simplified from of ${[{(\sqrt[5]{x^{- \frac{3}{5}}})}^{- \frac{5}{3}}]}^{- 5} = ?$

A. x⁵

B. x^-5

C. x

D. $\frac{1}{x}$

Discuss

Solution:

\begin{aligned} {[{(\sqrt[5]{x^{- \frac{3}{5}}})}^{- \frac{5}{3}}]}^{- 5} \\ = {[{(x^{- \frac{3}{25}})}^{- \frac{5}{3}}]}^{- 5} \\ = {[(x^{\frac{1}{5}})]}^{- 5} \\ = x^{- \frac{1}{5} \times 5} \\ = x^{- 1} \\ = \frac{1}{x} \end{aligned}

79. Find the value of x in the expression : $\sqrt[4]{3 x + 1} = 2$

A. 3

B. 6

C. 4

D. 5

Discuss

Solution:

\begin{aligned} \sqrt[4]{3 x + 1} = 2 \\ \Rightarrow {(\sqrt[4]{3 x + 1})}^{4} = 2^{4} \\ \Rightarrow {(3 x + 1)}^{4 \times \frac{1}{4}} = 16 \\ \Rightarrow 3 x = 15 \\ \Rightarrow x = 5 \end{aligned}

80. (0.04)² ÷ (0.008) × (0.2)⁶ = (0.2)^?

A. 5

B. 6

C. 8

D. 9

Discuss

Solution:

\begin{aligned} Let {(0.04)}^{2} \div (0.008) \times {(0.2)}^{6} = {(0.2)}^{x} \\ Then, {(0.2)}^{x} = {[{(0.2)}^{2}]}^{2} \div {(0.2)}^{3} \times {(0.2)}^{6} \\ \Leftrightarrow {(0.2)}^{x} = {(0.2)}^{(2 \times 2)} \div {(0.2)}^{3} \times {(0.2)}^{6} \\ \Leftrightarrow {(0.2)}^{x} = {(0.2)}^{4} \div {(0.2)}^{3} \times {(0.2)}^{6} \\ \Leftrightarrow {(0.2)}^{x} = {(0.2)}^{(4 - 3 + 6)} \\ \Leftrightarrow {(0.2)}^{x} = {(0.2)}^{7} \\ \Leftrightarrow x = 7 \end{aligned}

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