Mathematics · Quantitative Aptitude

Surds and Indices

408 Questions

Surds and indices questions focus on evaluating square roots, cube roots, and fractional exponents. These topics are a core part of the quantitative aptitude section in many competitive exams. Practicing these problems builds speed and accuracy for solving numerical equations.

Square roots evaluationCube roots calculationExponents and powersFractional exponentsSurds multiplication

Surds and Indices Questions

Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

Find the value of cube root of the number $2486$. (Round off your number to the nearest whole number)

  1. $11$
  2. $12$
  3. $13$
  4. $14$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We need to find value of $\sqrt[3]{2486}$
Take, $n = 2486$, choose any starting value of $x$.
So, $13^3$ is $2197 < 2486$
So, $x = 13$
$x _\text{next} =$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
$x _\text{next} =$ $\dfrac{2}{3}13+\dfrac{2486}{3\times 13^2}$
$x _\text{next} = 13.56$
So, the nearest whole number for the cube root $2486$ is $14$.

Multiple choice maths multiply and divide division trick division division of numbers

Solve it 
$\dfrac {\left( {{{\left( {245 + 232} \right)}^2} - {{\left( {245 - 232} \right)}^2}} \right)}{\left( {245 + 232} \right)}$

  1. $4$
  2. $2$
  3. $232$

  4. none of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$=\dfrac{{\left(245+232\right)}^{2}-{\left(245-232\right)}^{2}}{\left(245+232\right)}$
$=\dfrac{\left(245+232-245+232\right)\left(245+232+245-232\right)}{\left(245+232\right)}$
$=\dfrac{2\left(232\right)2\left(245\right)}{\left(245+232\right)}$
$=\dfrac{2,27,360‬}{477}=476.65$
Multiple choice maths surface area and volume of cube and cuboid finding out the diagonal of cube and cuboid length of the diagonal of cube diagonal of cube and cuboid

Find the pythagorean triplet.

  1. $8, 15, 17$
  2. $9, 10, 15$
  3. $9, 10, 17$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Every right triangle has side length satisfying:

a$^{2}$ $+$ b$^{2}$ $=$ c$^{2}$
c is the longest side,
Here 
$8^{2}$ $+$ $15^{2}$ $=$ $17^{2}$
Hence it is a pythagorean triplet.
Option A is correct.

Multiple choice maths number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

The greater number between $\sqrt{17}-\sqrt{12}$ and $\sqrt{11}-\sqrt{6}$ is ____.

  1. $\sqrt{17}-\sqrt{12}$
  2. $\sqrt{11}-\sqrt{6}$

  3. Both are equal

  4. Cannot comare
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt{17}=4.12\ \sqrt {12}=3.46\ \therefore\sqrt{17}-\sqrt{12}=0.66$

$\sqrt{11}=3.32\ \sqrt6=2.45\ \therefore\sqrt{11}-\sqrt6=0.87$

$\sqrt{11}-\sqrt6>\sqrt{17}-\sqrt{12}$

Multiple choice maths indices negative indices law of indices laws of indices

$\left { \left (\dfrac {3}{4}\right )^{-1} - \left (\dfrac {1}{4}\right )^{-1}\right }^{-1} = ?$

  1. $\dfrac {3}{8}$
  2. $\dfrac {-3}{8}$
  3. $\dfrac {8}{3}$
  4. $\dfrac {-8}{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We need to find value of $\left { \left (\dfrac {3}{4}\right )^{-1} - \left (\dfrac {1}{4}\right )^{-1}\right }^{-1} $
It can be written as $\left (\dfrac {4}{3} - 4\right)^{-1}$ $=$ $\left (\dfrac {-8}{3}\right)^{-1}$ $=$ $-\dfrac {3}{8}$

Multiple choice maths indices negative indices law of indices laws of indices

$\left {\left (\dfrac {1}{3}\right )^{-3} -\left (\dfrac {1}{2}\right )^{-3}\right } \div \left (\dfrac {1}{4}\right )^{-3} = ?$

  1. $\dfrac {19}{64}$
  2. $\dfrac {64}{19}$
  3. $\dfrac {27}{16}$
  4. $\dfrac {16}{27}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
We need to find value of $\left \{\left (\dfrac {1}{3}\right )^{-3} -\left (\dfrac {1}{2}\right )^{-3}\right \} \div \left (\dfrac {1}{4}\right )^{-3} = ?$
$\left (\dfrac{1}{3}\right)^{-3}=3^{3}$
$\left (\dfrac{1}{2}\right)^{-3}=2^{3}$
$\left (\dfrac{1}{4}\right)^{-3}=4^{3}$
So, given expression becomes,
$\dfrac {3^{3}-2^{3}}{4^{3}}$ $=\dfrac {19}{64}$. 
Hence, A is the right option.
Multiple choice maths indices negative indices law of indices laws of indices

If $\sqrt [ 3 ]{ a+\sqrt { b }  } =7+4\sqrt { 3 } $, then $\sqrt [ 3 ]{ { a }^{ 2 }-b } =$

  1. $0$
  2. $1$
  3. $-1$
  4. $7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have.

$ \sqrt[3]{a+\sqrt{b}}=7+4\sqrt{3} $

$ {{(a+\sqrt{b})}^{1/3}}=(7+4\sqrt{3}) $

$ a+\sqrt{b}={{(7+4\sqrt{3})}^{3}}\ \ ......\ \ (1) $


$ \text{Similarly,} $

$ a-\sqrt{b}={{(7-4\sqrt{3})}^{3}}\ \ ......\ \ (2) $


On multiplying (1) and (2) to. We get,

$ (a+\sqrt{b})(a-\sqrt{b})={{(7+4\sqrt{3})}^{3}}{{(7-4\sqrt{3})}^{3}} $

$ {{a}^{2}}-b={{\left[ (7+4\sqrt{3})(7-4\sqrt{3}) \right]}^{3}} $

$ {{a}^{2}}-b={{\left[ 49-16\times 3 \right]}^{3}} $

$ {{({{a}^{2}}-b)}^{1/3}}=(1) $

$ \sqrt[3]{{{a}^{2}}-b}=1 $


Hence, this is the answer

Multiple choice maths indices negative indices law of indices laws of indices

The value of ${ \left[ { \left( { 3 }^{ 2 } \right)  }^{ 2 } \right]  }^{ -1 }$ is-

  1. $81$
  2. $-81$
  3. $-0.0123$
  4. $0.0123$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$\cfrac { 1 }{ { \left( { 3 }^{ 2 } \right)  }^{ 2 } } $        $\because (a^m)^n=a^{mn}$
$=\cfrac { 1 }{ { 3 }^{ 4 } }$
$ =\cfrac { 1 }{ 81 }$
$ =0.0123$
Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

Arrange in ascending order of magnitude $\sqrt 3, \sqrt [5]{15}, \sqrt [10]{227}$

  1. $\sqrt [5]{15} < \sqrt [10]{227} < \sqrt 3$
  2. $\sqrt 3 < \sqrt [5]{15} < \sqrt [10]{227}$
  3. $\sqrt [10]{227} < \sqrt 3 < \sqrt [5]{15}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt 3, \sqrt [5]{15}, \sqrt [10]{227}$


LCM of $2, 5$ and $10=10$

$\sqrt 3=\sqrt [2\times 5]{3^5}=\sqrt [10]{3\times 3\times 3\times 3\times 3}=\sqrt [10]{243}$

$\sqrt [5]{15}=\sqrt [5\times 2]{15^2}=\sqrt [10]{15\times 15}=\sqrt [10]{225}$

$\sqrt [10]{227}=\sqrt [10]{227}$

$\therefore \sqrt [5]{15} < \sqrt [10]{227} < \sqrt 3$

Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

Arrange in ascending order $\sqrt [ 6 ]{ 7 } ,\sqrt [ 4 ]{ 3 } ,\sqrt [ 12 ]{ 48 } $

  1. $\sqrt [ 4 ]{ 3 } ,\sqrt [ 12 ]{ 48 } ,\sqrt [ 6 ]{ 7 } $
  2. $\sqrt [ 12 ]{ 48 } ,\sqrt [ 4 ]{ 3 } ,\sqrt [ 6 ]{ 7 } $
  3. $\sqrt [ 6]{ 7 } ,\sqrt [ 12 ]{ 48 } ,\sqrt [ 4 ]{ 3 } $
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Convert to powers: 7^(1/6) = 7^(2/12) = 49^(1/12); 3^(1/4) = 3^(3/12) = 27^(1/12); 48^(1/12). Comparing the bases: 27 < 48 < 49. Thus, 3^(1/4) < 48^(1/12) < 7^(1/6).

Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

The ascending order of $\sqrt { 2 } ,\sqrt [ 3 ]{ 4 } ,\sqrt [ 4 ]{ 6 } $ is

  1. $\sqrt { 2 } ,\sqrt [ 3 ]{ 4 } ,\sqrt [ 4 ]{ 6 } $
  2. $\sqrt { 2 } ,\sqrt [ 4 ]{ 6 } ,\sqrt [ 3 ]{ 4 } $
  3. $\sqrt [ 3 ]{ 4 }, \sqrt {2},\sqrt [ 4 ]{ 6 } $
  4. $\sqrt [ 4 ]{ 6 },\sqrt [ 3 ]{ 4 } ,\sqrt {2}$
Reveal answer Fill a bubble to check yourself
A Correct answer