Practice MCQ Questions and Answer on Square Root and Cube Root

71.

If $$\sqrt {24} = 4.889,$$   the value of $$\sqrt {\frac{8}{3}} $$   is = ?

• (A) 0.544
• (B) 1.333
• (C) 1.633
• (D) 2.666

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 $$\eqalign{ & = \sqrt {\frac{8}{3}} \cr & = \sqrt {\frac{{8 \times 3}}{{3 \times 3}}} \cr & = \frac{{\sqrt {24} }}{3} \cr & = \frac{{4.899}}{3} \cr & = 1.633 \cr} $$

72.

The number $${\text{2}}{{\text{5}}^{64}} \times {64^{25}}$$   is the square of a natural number n. The sum of the digits of n is = ?

• (A) 7
• (B) 14
• (C) 21
• (D) 28

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 $$\eqalign{ & \Leftrightarrow {{\text{n}}^2} = {\left( {{\text{25}}} \right)^{64}} \times {\left( {64} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {\left( {{{\text{5}}^2}} \right)^{64}} \times {\left( {{2^6}} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{150}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{128}} \times {2^{22}} \cr & \Leftrightarrow n = {5^{64}} \times {2^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {\left( {5 \times 2} \right)^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {10^{64}} \times 2048 \cr} $$ ∴ Sum of digits of n = 2 + 0 + 4 + 8 = 14

73.

What percentage of the numbers from 1 to 50 have squares that end in the digit 1 ?

• (A) 1%
• (B) 5%
• (C) 10%
• (D) 11%

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Answer & Solution Answer: Option E Solution: The squares of numbers having 1 and 9 as the unit's digit end in the digit 1. $$\eqalign{ & {\text{Such numbers are,}} \cr & 1,9,11,19,21,29,31,39,41,49{\text{ i}}{\text{.e}}{\text{.,}} \cr & {\text{There are 10 such numbers}}{\text{.}} \cr & \therefore {\text{Required percentage}} \cr & = \left( {\frac{{10}}{{50}} \times 100} \right)\% \cr & = 20\% \cr} $$

74.

If $$\sqrt 5 = 2.236, $$   then the value of $$\frac{{\sqrt 5 }}{2}$$ $$ - $$ $$\frac{{10}}{{\sqrt 5 }}$$ $$ + $$ $$\sqrt {125} $$   is equal to :

• (A) 5.59
• (B) 7.826
• (C) 8.944
• (D) 10.062

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 $$\eqalign{ & \frac{{\sqrt 5 }}{2} - \frac{{10}}{{\sqrt 5 }} + \sqrt {125} \, \cr & = \frac{{{{\left( {\sqrt 5 } \right)}^2} - 20 + 2\sqrt 5 \times 5\sqrt 5 }}{{2\sqrt 5 }} \cr & = \frac{{5 - 20 + 50}}{{2\sqrt 5 }} \cr & = \frac{{35}}{{2\sqrt 5 }} \times \frac{{\sqrt 5 }}{{\sqrt 5 }} \cr & = \frac{{35\sqrt 5 }}{{10}} \cr & = \frac{{7 \times 2.236}}{2} \cr & = 7 \times 1.118 \cr & = 7.826 \cr} $$

75.

For what value of * the statement $$\left( {\frac{*}{{15}}} \right)$$ $$\left( {\frac{*}{{135}}} \right)$$  = 1 is true ?

• (A) 15
• (B) 25
• (C) 35
• (D) 45

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 $$\eqalign{ & {\text{Method 1:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow x = \sqrt {15 \times \left(15 \times 9 \right)} \cr & \Leftrightarrow x = \sqrt {{{15}^2} \times {3^2}} \cr & \Leftrightarrow x = 15 \times 3 \cr & \Leftrightarrow x = 45 \cr & \cr & {\text{Method 2:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow {x^2} = 2025 \cr & \Leftrightarrow x = \sqrt {{2025} } \cr & \Leftrightarrow x = 45 \cr} $$

76.

$${\left( {15} \right)^2} + {\left( {18} \right)^2} - 20 = \sqrt ? $$

• (A) 22
• (B) 23
• (C) 529
• (D) 279841

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 $$\eqalign{ & {\text{Let, }} \cr & {\left( {15} \right)^2} + {\left( {18} \right)^2} - 20 = \sqrt x \cr & {\text{Then, }} \cr & \sqrt x = 225 + 324 - 20 \cr & \Leftrightarrow x = {\left( {529} \right)^2} \cr & \Leftrightarrow x = 279841 \cr} $$

77.

What should come in place of both x in the equation $$\frac{x}{{\sqrt {128} }} = \frac{{\sqrt {162} }}{x}$$

• (A) 12
• (B) 14
• (C) 144
• (D) 196

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 $$\eqalign{ & {\text{Let}}\,\frac{x}{{\sqrt {128} }} = \frac{{\sqrt {162} }}{x} \cr & {\text{Then}}\,{x^2} = \sqrt {128 \times 162} \cr & = \sqrt {64 \times 2 \times 18 \times 9} \cr & = \sqrt {{8^2} \times {6^2} \times {3^2}} \cr & = 8 \times 6 \times 3 \cr & = 144 \cr & \therefore x = \sqrt {144} = 12 \cr} $$

78.

The smallest number to be added to 680621 to make the sum a perfect square is = ?

• (A) 4
• (B) 5
• (C) 6
• (D) 8

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 $$\eqalign{ & \,\,\,\,\,\,\,\,8|\overline {68} \,\,\overline {06} \,\,\overline {21} \,(824 \cr & \,\,\,\,\,\,\,\,\,\,\,\,|\,\,64 \cr & \,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,162|\,\,\,\,4\,06 \cr & \,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,3\,24 \cr & \,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & 1644|\,\,\,\,\,\,\,\,\,\,82\,21 \cr & \,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,65\,76 \cr & \,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,16\,45 \cr & \therefore {\text{Number to be added}} \cr & = {\left( {825} \right)^2} - 680621 \cr & = 680625 - 680621 \cr & = 4 \cr} $$

79.

The number of digits in the square root of 625685746009 is = ?

• (A) 4
• (B) 5
• (C) 6
• (D) 7

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 The number of digits of the square root of a perfect square number of n digits is $$\eqalign{ & {\text{(i)}}\frac{n}{2}{\text{, if n is even}} \cr & {\text{(ii)}}\frac{{n + 1}}{2}{\text{, if n is odd}} \cr & {\text{Here, }}n = 12 \cr & {\text{So, required number of digits}} \cr & = \frac{n}{2} \cr & = \frac{{12}}{2} \cr & = 6{\text{ }} \cr} $$

80.

$${1.5^2} \times \sqrt {0.0225} = ?$$

• (A) 0.0375
• (B) 0.3375
• (C) 3.275
• (D) 32.75

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 $$\eqalign{ & = {1.5^2} \times \sqrt {0.0225} \cr & = {1.5^2} \times \sqrt {\frac{{225}}{{10000}}} \cr & = 2.25 \times \frac{{15}}{{100}} \cr & = 2.25 \times 0.15 \cr & = 0.3375 \cr} $$