Mathematics · Quantitative Aptitude

Surds and Indices

362 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 comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Compare the following pairs of surds. $\sqrt[4]{64}, \sqrt[6]{128}$    

  1. $\sqrt[4]{64} > \sqrt[6]{128}$
  2. $\sqrt[4]{64} < \sqrt[6]{128}$
  3. $\sqrt[4]{64} \neq \sqrt[6]{128}$
  4. $\sqrt[4]{64} = \sqrt[6]{128}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt[4]{64}=\sqrt[4]{2^6}=2\sqrt[4]{2^2}=2\sqrt{2}=2\sqrt[6]{2^3}=2\sqrt[6]{8}$
$\sqrt[6]{128}=\sqrt[6]{2^7}=2\sqrt[6]{2}$
$\sqrt[4]{64}>\sqrt[6]{128}$

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

$\sqrt{11}-\sqrt{10} .... \sqrt{12}-\sqrt{11}$,use appropriate inequality to fill the gap.

  1. <

  2. >
  3. $=$

  4. cannot determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We first consider $\sqrt { 11 } -\sqrt { 10 }$ as follows:


$\sqrt { 11 } -\sqrt { 10 } =3.317-3.162=0.156$

Now we find the value of $\sqrt { 12 } -\sqrt { 11 }$ as follows:

$\sqrt { 12 } -\sqrt { 11 } =3.464-3.317=0.147$


Since $0.156>0.147$

Hence, $\sqrt { 11 } -\sqrt { 10 }>\sqrt {12} -\sqrt {11}$

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Arrange the following in ascending order of magnitude: $\displaystyle \sqrt[4]{90}, \sqrt[3]{10}, \sqrt{6}$

  1. $\displaystyle \sqrt{3} < \sqrt[4]{10} < \sqrt[3]{6}$
  2. $\displaystyle \sqrt{3} > \sqrt[4]{10} > \sqrt[3]{6}$
  3. $\displaystyle \sqrt{3} > \sqrt[4]{10} < \sqrt[3]{6}$
  4. $\displaystyle \sqrt{3} < \sqrt[4]{10} > \sqrt[3]{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Arrange the following surds in ascending order of their magnitudes: $\sqrt{5},\sqrt [ 3 ]{ 11 } ,2\sqrt [ 6 ]{ 3 } $

  1. $\sqrt [ 3 ]{ 11 } > \sqrt{5}< 2\sqrt [ 6 ]{ 3 } $
  2. $\sqrt [ 3 ]{ 11 } < \sqrt{5}< 2\sqrt [ 6 ]{ 3 } $
  3. $\sqrt [ 3 ]{ 11 } > \sqrt{5}> 2\sqrt [ 6 ]{ 3 } $
  4. $\sqrt [ 3 ]{ 11 } < \sqrt{5}> 2\sqrt [ 6 ]{ 3 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt{5} = 5^{1/2}$

$\sqrt[3]{11} = 11^{1/3}$

$2\sqrt[6]{3} = \sqrt[6]{12}= 12^{1/6}$

L.C.M of the denominators of the exponents is 12.

So,

$\sqrt{5} = 5^{\frac{1}{2}\times\frac{6}{6}} = \sqrt [12]{5^6}=\sqrt[12]{15625}$

$\sqrt[3]{11} = 11^{\frac{1}{3}\times\frac{4}{4}} = \sqrt [12]{11^4} =\sqrt[12]{14641}$

$2\sqrt[6]{3} = \sqrt[6]{12}= 12^{\frac{1}{6}\times\frac{2}{2}} = \sqrt[12]{12^2} =\sqrt[12]{144}$

Hence, the Ascending order is $2\sqrt[6]{3}, \sqrt[3]{11},\sqrt{5}$

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Which is greater?
${ \left( \cfrac { 1 }{ 2 }  \right)  }^{ 1/2 } $ or ${ \left( \cfrac { 2 }{ 3 }  \right)  }^{ 1/3 } $

  1. ${ \left( \cfrac { 2 }{ 3 } \right) }^{ 1/3 } $
  2. ${ \left( \cfrac { 1 }{ 2 } \right) }^{ 1/2 } $

  3. Both are equal

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

$ \left(\dfrac{1}{2}\right)^{1/2} \; or \; \left(\dfrac{2}{3}\right)^{1/3}$


$= \left(\left(\dfrac{1}{2}\right)^{1/2}\right)^6 \; or \; \left(\left(\dfrac{2}{3}\right)^{1/3}\right)^6$


$= \left(\dfrac{1}{2}\right)^3 \; or \; \left(\dfrac{2}{3}\right)^2$


$= \left(\dfrac{1}{8}\right) \; or \; \left(\dfrac{4}{9}\right)$


= 0.125 or 0.44


Since, 0.44 is greater and so is $ \left(\dfrac{2}{3}\right)^{1/3}$

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

The correct descending order of the following surds is 

$ \sqrt [3]{2}$, $\sqrt 3$, $\sqrt 4$, $\sqrt 5$

  1. $\sqrt 3$ > $\sqrt 4$ > $\sqrt 5$ > $\sqrt [3]{2}$
  2. $\sqrt 4$ > $\sqrt 5$ > $\sqrt [3]{2}$ > $\sqrt 3$
  3. $\sqrt 5$ > $\sqrt 4$ > $\sqrt 3$ > $\sqrt [3]{2}$
  4. $\sqrt [3]{2}$ > $\sqrt 4$ > $\sqrt 3$ > $\sqrt 5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

LCM of $3$ and $2$ is $6$.
Therefore, multiplying the index of all numbers by $6$.
${\sqrt [3]{2}}^6 = 2^\dfrac 63 = 2^2 = 4$

${\sqrt 3}^6 = 3^\dfrac 62 = 3^3 = 27$

${\sqrt 4}^6 = 4^ \dfrac 62 = 4^3 = 64$

${\sqrt 5}^6 = 5^{\dfrac 62} =  5^3 = 125$

$\therefore 125 > 64 > 27 > 4$

So, option $C$ is correct.

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The triplicate ratio of $\sqrt [3]{9} : \sqrt {8}$ is ____

  1. $9 : 16$
  2. $9 : 16\sqrt {2}$
  3. $9 : 8$
  4. $3 : 2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt [3]{9} : \sqrt {8} = (9)^{\dfrac {1}{3}} : (8)^{\dfrac {1}{2}}$
The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\sqrt [3]{9} : \sqrt {8}$ is $[(9)^{\dfrac {1}{3}}]^{3} : [(8)^{\dfrac {1}{2}}]^{3}$
$= 9 : (2\sqrt {2})^{3}$
$= 9 : 8(\sqrt {2})^{3}$
$= 9 : 16\sqrt {2}$.

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The triplicate ratio of $\sqrt [3]{b^{2}} : \sqrt [3]{a^{2}}$ is _____

  1. $b^{2} : a^{2}$
  2. $a : b$
  3. $b^{3} : a^{3}$
  4. $a^{2} : b^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\sqrt [3]{b^{2}} : \sqrt [3]{a^{2}}$ is $(\sqrt [3]{b^{2}})^{3} : (\sqrt [3]{a^{2}})^{3} = b^{2} : a^{2}$.

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The subtriplicate ratio of $\sqrt {x} : \sqrt {y}$ is ____

  1. $x^{\frac {1}{9}} : y^{\frac {1}{9}}$
  2. $\sqrt {y} : \sqrt {x}$
  3. $x^{\frac {1}{3}} : y^{\frac {1}{6}}$
  4. $x^{\frac {1}{6}} : y^{\frac {1}{6}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $\sqrt {x} : \sqrt {y} = x^{\cfrac {1}{2}} : y^{\frac {1}{2}}$ is $(x^{\cfrac {1}{2}})^{\cfrac {1}{3}} : (y^{\cfrac {1}{2}})^{\cfrac {1}{3}}$
$= x^{\cfrac {1}{6}} : y^{\cfrac {1}{6}}$

Multiple choice

√50  + √32

  1. 8√2

  2. 9√2

  3. 10√3

  4. 7√5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

sqrt(50) = 5*sqrt(2) and sqrt(32) = 4*sqrt(2). Adding them gives 5*sqrt(2) + 4*sqrt(2) = 9*sqrt(2).

Multiple choice

√27 + √75

  1. 7√3

  2. 6√3

  3. 8√3

  4. 9√3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

sqrt(27) = 3*sqrt(3) and sqrt(75) = 5*sqrt(3). Adding them gives 3*sqrt(3) + 5*sqrt(3) = 8*sqrt(3).

Multiple choice

√250 - √160

  1. 2√10

  2. 5√4

  3. √10

  4. √5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

sqrt(250) = 5*sqrt(10) and sqrt(160) = 4*sqrt(10). Subtracting them gives 5*sqrt(10) - 4*sqrt(10) = sqrt(10).

Multiple choice

√80 - √5

  1. 4√5

  2. 5√5

  3. 6√5

  4. 3√5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

sqrt(80) = 4*sqrt(5). Subtracting sqrt(5) gives 4*sqrt(5) - sqrt(5) = 3*sqrt(5).