Power, Indices and Surds Questions for Competitive Exams

Power, Indices and Surds Questions for Competitive Exams. Important MCQ, selected from the previous year exam questions papers of SSC CGL, CPO, CHSL, Bank, UPSSSC and other govt jobs examinations for practice. topic wise question and answer of Power, Indices and Surds with solutions are very useful for upcoming competitive Examinations .

Power, Indices and Surds Questions

Finding the Largest and Smallest Values

Q.1: The greatest number among 3⁵⁰, 4⁴⁰, 5³⁰ and 6²⁰ is
a) 3⁵⁰
b) 4⁴⁰
c) 5³⁰
d) 6²⁰

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Ans : b) 4⁴⁰
3⁵⁰ =(3⁵)¹⁰ = (243)¹⁰
4⁴⁰ = (4⁴)¹⁰=(256)¹⁰
5³⁰ = (5³)¹⁰ = (125)¹⁰
6²⁰ = (6²)¹⁰ = (36)¹⁰

Q.2: The greatest number among the following is $\frac 49, \sqrt{\frac {9}{49}}, 0.47, (0.7)^2$
a) $\frac 49$
b) $\sqrt{\frac {9}{49}}$
c) 0.47
d) $(0.7)^2$

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Ans : d) $(0.7)^2$
$\frac 49$ =0.44
$\sqrt{\frac {9}{49}} = \frac37 = 0.43$
$(0.7)^2 = 0.49$

Q.3: Arranging the following in descending order:
$\sqrt[3]{4}, \sqrt 2, \sqrt[6]{3}, \sqrt[4]{5},$
a) $\sqrt[3]{4} > \sqrt[4]{5} > \sqrt 2 > \sqrt[6]{3}$
b) $\sqrt[3]{4} > \sqrt 2 > \sqrt[6]{3}>\sqrt[4]{5}$
c) $\sqrt 2 >\sqrt[3]{4} > \sqrt[6]{3}> \sqrt[4]{5}$
d) $\sqrt[6]{3} >\sqrt[4]{5} >\sqrt[3]{4}> \sqrt 2$

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Ans : a) $\sqrt[3]{4} > \sqrt[4]{5} > \sqrt 2 > \sqrt[6]{3}$
$\sqrt[3]{4} = 4^\frac13 = (4^4) = (256)^\frac {1}{12}$
$\sqrt 2 = 2^\frac{1}{12} =(64)^\frac{1}{12}$
$\sqrt [6]3 = 3^\frac{1}{6} = (3^2)^\frac {1}{12} = (9)^\frac {1}{12}$
$\sqrt[4] 5 =5^{\frac14}= (5^3)^{ \frac{1}{12}} =(125)^\frac{1}{12}$

Q.4: The smallest among the numbers 2²⁵⁰, 3¹⁵⁰, 5¹⁰⁰, 4²⁰⁰ is
a) 2²⁵⁰
b) 3¹⁵⁰
c) 5¹⁰⁰
d) 4²⁰⁰

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Ans : c) 5¹⁰⁰
2²⁵⁰ =(2⁵)⁵⁰ =(32)⁵⁰
3¹⁵⁰ = (3³)⁵⁰ = (27)⁵⁰
5¹⁰⁰ = (5²)⁵⁰ =(25)⁵⁰
4²⁰⁰ = (4⁴)⁵⁰= (256)⁵⁰

Q.5: Which is greater $\sqrt[3]2$ or $\sqrt3$ ?
a) $\sqrt[3]2$
b) $\sqrt3$
c) Equal
d) Can not be compared

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Ans : $\sqrt3$
Cube of both the numbers are
$(\sqrt[3]2)^3 =2 \: \text {and} \: (\sqrt3)^3 = 3 \sqrt3$

Q.6: The smallest among $\sqrt[6]{12}, \sqrt[3]4, \sqrt[4]5, \sqrt3$ is
a) $\sqrt[6]{12}$
b) $\sqrt[3]4$
c) $\sqrt[4]5$
d) $\sqrt3$

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Ans : c) $\sqrt[4]5$
LCM of 2,3,4 and 6 is 12
$\sqrt[6]{12}$ =(12)^1/6 =(12)^2/12 = (12²)^1/12 = (144)^1/12
$\sqrt[3]4$ = (256)^1/12
$\sqrt[4]5$ = (125)^1/12
$\sqrt3$ = (729)^1/12

Q.7: If X =(0.25)1\2, Y = (0.4)2, Z=(0.216)1/3, then
a) Y>X>Z
b) X>Y>z
c) Z>X>Y
d) X>Z>Y

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Ans : c) Z>X>Y
X=(0.25)1/2 =0.5
Y= (0.4)2 = 0.16
Z = (0.216) = 0.6

Simplifying when the Root Values are given

Q.8: If $\sqrt {33} =5.745, \text {than the value of} \sqrt {\frac {3}{11}}$ is approximately.
a) 1
b) 0.5223
c) 6.32
d) 2.035

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Ans : b) 0.5223
$\sqrt {\frac {3}{11}} = \sqrt {\frac {3\times 11}{11 \times 11}} = \frac {1}{11} \sqrt {33} = \frac{5.745}{11} = 0.5223$

Q.9: If $\sqrt 2 = 1.4142......$ is given, then the value of $\dfrac {7} {(3+\sqrt2)}$ correct up to two decimal places is :
a) 1.59
b) 1.60
c) 2.58
d) 2.57

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Ans : a) 1.59
$\frac {7} {(3+\sqrt2)} = \frac {7} {(3+\sqrt2)} \times \frac {3-\sqrt2}{3-\sqrt2} = \frac {21-7\sqrt2}{9-2} =3-\sqrt2 = 3-1.4142 =1.59$

Q.10: Evaluate : $16\sqrt {\frac34} - 9\sqrt \frac 34 \: \: \: \text {if} \: \sqrt {12} = 3.46$
a) 3.46
b) 10.38
c) 13.84
d) 24.22

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Ans : a) 3.46
$16\sqrt {\frac{3\times 4} {4\times4} } - 9\sqrt {\frac {4\times3}{3\times 3}}$
= $\frac {16}{4}\sqrt{12}-\frac 93 \sqrt {12}$
= $4 \sqrt{12} -3\sqrt{12} = \sqrt {12} = 3.46$

Rationalising or Prime Factor

Q.11: The number of prime factors in 6³³³ x 7²²² x 8¹¹¹
a) 1221
b) 1222
c) 1111
d) 1211

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Ans : a) 1221
6³³³ x 7²²² x 8¹¹¹ = (2×3)³³³ x 7²²² x (2³)¹¹¹ = 2³³³ x 3³³³ x 7²²² x 2³³³
Total Prime Factor = 333+333+222+333= 1221

Q.12: The total number of prime factors in 4¹⁰ x 7³ x 16² x 11 x 10² is
a) 34
b) 35
c) 36
d) 37

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Ans : c) 36
4¹⁰ x 7³ x 16² x 11 x 10²
= (2²)¹⁰ x 7³ x (2⁴)² x 11 x (5×2)²
= 2²⁰ x 7³ x 2⁸ x 11 x 5² x2²
=2³⁰ x 5² x 7³ x 11
Total Prime Factors = 30+2+3+1 = 36

Q.13: The rationalizing factor of $3 \sqrt3$ is
a) $\frac 13$
b) 3
c) -3
d) $\sqrt 3$

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Ans : d) $\sqrt 3$
$3 \sqrt3 \times \sqrt3$ = 3 x 3 =9

Positive and Negative Exponent

Q.14: The quotient when 10¹⁰⁰ is divided by 5⁷⁵ is
a) 2²⁵ x 10⁷⁵
b) 10²⁵
c) 2⁷⁵
d) 2⁷⁵ x 10²⁵

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Ans : d) 2⁷⁵ x 10²⁵

Q.15: If 3^x+8= 27^2x+1 , then the value of x is :
a) 7
b) 3
c) -2
d) 1

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Ans : d) 1
3^x+8= 27^2x+1 = (3³)^2x+1 =3^6x+3
x+8 =6x + 3
5x = 5, x=1

Q.16: (36)^1/6 is equal to
a) 1
b) 6
c) $\sqrt 6$
d) $\sqrt[3]{6}$

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Ans : $\sqrt[3]{6}$
(36)^1/6 = (6²)^1/6 = 6^1/3 = $\sqrt[3]{6}$

Based on Square Root Series : Power, Indices and Surds Questions

Q.17: The value of the expression is :
$\sqrt {6+\sqrt {6+\sqrt{6+ ...........}}}$
a) 5
b) 3
c) 2
d) 30

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Ans : b) 3
Let $\sqrt {6+\sqrt {6+\sqrt{6+ ...........}}} = x$
Squaring both side
$6 + \sqrt {6+\sqrt {6+\sqrt{6+ ...........}}} = x^2$
= $x^2 = 6+x$
= $x^2 - 3x +2x -6 = 0$
= $(x-3)(x-2) = 0$
= $x = 3$

Q.18: $\dfrac {\sqrt{10+ \sqrt{25+ \sqrt{108+ \sqrt{154+ \sqrt{225}}}}}} {\sqrt[3]8}$ = ?
a) 4
b) 2
c) 8
d) $\frac12$

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Ans : b) 2
$\dfrac {\sqrt{10+ \sqrt{25+ \sqrt{108+ \sqrt{154+ 15}}}}} {2}$

= $\dfrac {\sqrt{10+ \sqrt{25+ \sqrt{108+ \sqrt{169}}}}} {2}$

= $\dfrac {\sqrt{10+ \sqrt{25+ \sqrt{108+ 13}}}} {2}$

= $\dfrac {\sqrt{10+ \sqrt{25+ 11}}} {2}$

= $\dfrac {\sqrt{10+ 6}} {2} = \dfrac 42 = 2$

Power, Indices and Surds Questions in Hindi – NRA STUDY

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