Practice MCQ Questions and Answer on Surds and Indices

91.

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

Show Answer

 $$\eqalign{ & 16\sqrt {\frac{3}{4}} - 9\sqrt {\frac{4}{3}} \cr & \Rightarrow 16\sqrt {\frac{{3 \times 4}}{{4 \times 4}}} - 9\sqrt {\frac{{3 \times 4}}{{3 \times 3}}} \cr & \Rightarrow 16 \times \frac{{\sqrt {12} }}{4} - \frac{{9\sqrt {12} }}{3} \cr & \Rightarrow 4\sqrt {12} - 3\sqrt {12} \cr & \Rightarrow \sqrt {12} \cr & \Rightarrow 3.46 \cr} $$

92.

If x = $$\sqrt{a\sqrt[3]{ab\sqrt{a\sqrt[3]{ab}}}}.....\propto ,$$     then the value of x is:

• (A) $$\sqrt[5]{a^{2}b}$$
• (B) $$\sqrt[5]{a^4{b}^4}$$
• (C) $$\sqrt[6]{a^{5}b}$$
• (D) $$\sqrt[5]{a^{4}b}$$

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 $$\eqalign{ & x = \sqrt {a\root 3 \of {ab\sqrt {a\root 3 \of {ab} } } } ..... \propto \cr & {\text{Square both side}} \cr & {x^2} = a\,\root 3 \of {ab\,x} \,\,\,\,\,\left( {\because \sqrt {a\root 3 \of {ab} } ..... \propto } \right) \cr & {\text{Again cube both sides}} \cr & {x^6} = {a^3}ab\,x \cr & {x^5} = {a^4}b \cr & x = \root 5 \of {{a^4}b} \cr} $$

93.

If $$N=\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}},$$    then what is the value of $$N+\frac{1}{N}?$$

• (A) 2√2
• (B) 5
• (C) 10
• (D) 13

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 $$\eqalign{ & N = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 + \sqrt 3 }} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 + \sqrt 3 }} \times \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 - \sqrt 3 }} \cr & = \frac{{7 + 3 - 2\sqrt {21} }}{4} \cr & = \frac{{10 - 2\sqrt {21} }}{4} \cr & = \frac{5}{2} - \frac{{\sqrt {21} }}{2} \cr & {\text{Similarly }}\frac{1}{N} = \frac{5}{2} + \frac{{\sqrt {21} }}{2} \cr & \therefore N + \frac{1}{N} \cr & = \frac{5}{2} - \frac{{\sqrt {21} }}{2} + \frac{5}{2} + \frac{{\sqrt {21} }}{2} \cr & = \frac{5}{2} + \frac{5}{2} \cr & = 5 \cr} $$

94.

Simplify : (3) × (3) = ?84

• (A) (27)3
• (B) (27)5
• (C) (729)2
• (D) (729)3

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

95.

The value of $$\sqrt{28+10\sqrt{3}}-\sqrt{7-4\sqrt{3}}$$      is closest to:

• (A) 7.2
• (B) 5.8
• (C) 6.1
• (D) 6.5

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 $$\eqalign{ & \sqrt {28 + 10\sqrt 3 } - \sqrt {7 - 4\sqrt 3 } \cr & = \sqrt {{{\left( {5 + \sqrt 3 } \right)}^2}} - \sqrt {{{\left( {2 - \sqrt 3 } \right)}^2}} \cr & = 5 + \sqrt 3 - 2 + \sqrt 3 \cr & = 3 + 2\sqrt 3 \cr & = 3 + 2 \times 1.73 \cr & = 3 + 3.46 \cr & = 6.5 \cr} $$

96.

The exponential form of $$\sqrt{\sqrt{2}\times \sqrt{3}}\text{ is = ?}$$

• (A) $${\text{6}}^{-\frac{1}{2}}$$
• (B) $${\text{6}}^{\frac{1}{2}}$$
• (C) $${\text{6}}^{\frac{1}{4}}$$
• (D) 6

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 $$\eqalign{ & {\text{The exponential form of }} \cr & \sqrt {\sqrt 2 \times \sqrt 3 } \cr & = \sqrt {{6^{\frac{1}{2}}}} \cr & = {\left( {{6^{\frac{1}{2}}}} \right)^{\frac{1}{2}}} \cr & = {6^{\frac{1}{4}}} \cr} $$

97.

If 847 × 385 × 675 × 3025 = 3 × 5 × 7 × 11 then the value of ab - cd isabcd

• (A) 4
• (B) 7
• (C) 5
• (D) 1

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 847 × 385 × 675 × 3025 = 3a × 5b × 7c × 11d LHS ⇒ 847 × 385 × 675 × 3025 = (7 × 11 × 11) × (5 × 7 × 11) × (3 × 3 × 3 × 5 × 5) × (5 × 5 × 11 × 11) = 33 × 55 × 72 × 115 = 3a × 5b × 7c × 11d Compare value a = 3; b = 5; c = 2; d = 5 ⇒ ab - cd = 3 × 5 - 2 × 5 = 15 - 10 = 5

98.

$$\frac{2^{n+4}-2\times 2^n}{2\times 2^{\left(n+3\right)}}+2^{-3}$$     is equal to = ?

• (A) 2(n+1)
• (B) $$\left(\frac{9}{8}-2^n\right)$$
• (C) $$\left(-2^{n+1}+\frac{1}{8}\right)$$
• (D) 1

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 $$\eqalign{ & \frac{{{2^{n + 4}} - 2 \times {2^n}}}{{2 \times {2^{\left( {n + 3} \right)}}}} + {2^{ - 3}} \cr & = \frac{{{2^{n + 4}} - {2^{n + 1}}}}{{{2^{\left( {n + 4} \right)}}}} + \frac{1}{{{2^3}}} \cr & = \frac{{{2^{n + 1}}\left( {{2^3} - 1} \right)}}{{{2^{\left( {n + 4} \right)}}}} + \frac{1}{{{2^3}}} \cr & = \frac{{{2^{n + 1}} \times 7}}{{{2^{n + 1}} \times {2^3}}} + \frac{1}{{{2^3}}} \cr & = \left( {\frac{7}{8} + \frac{1}{8}} \right) \cr & = \frac{8}{8} \cr & = 1 \cr} $$

99.

Which of the following is true? $$\begin{array}{rl}& \text{I}.\sqrt[3]{11}>\sqrt{7}>\sqrt[4]{45}\\ & \text{II}.\sqrt{7}>\sqrt[3]{11}>\sqrt[4]{45}\\ & \text{III}.\sqrt{7}>\sqrt[4]{45}>\sqrt[3]{11}\\ & \text{IV}.\sqrt[4]{45}>\sqrt{7}>\sqrt[3]{11}\end{array}$$

• (A) Only I
• (B) Only II
• (C) Only III
• (D) Only IV

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 $$\eqalign{ & \root 3 \of {11} ,\,\sqrt 7 ,\,\root 4 \of {45} \cr & {11^{\frac{1}{3}}},\,{7^{\frac{1}{2}}},\,{45^{\frac{1}{4}}} \cr & {\text{Take LCM of 3, 2, 4 is 12}} \cr & {11^4},\,{7^6},\,{45^3} \cr & {\text{14641,}}\,{\text{117649,}}\,{\text{91125}} \cr & 117649 > 91125 > 14641 \cr & {\text{Option C is true}} \cr & \sqrt 7 > \root 4 \of {45} > \root 3 \of {11} \cr} $$

100.

The value of $$\sqrt{40+\sqrt{9\sqrt{81}}}$$    is = ?

• (A) $$\sqrt{111}$$
• (B) 9
• (C) 7
• (D) 11

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 $$\eqalign{ & \sqrt {40 + \sqrt {9\sqrt {81} } } \cr & \Rightarrow \sqrt {40 + \sqrt {9 \times 9} } \cr & \Rightarrow \sqrt {40 + 9} \cr & \Rightarrow \sqrt {49} \cr & \Rightarrow 7 \cr} $$