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 maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\sqrt{10+ \sqrt{25+ \sqrt{x+ \sqrt{154+ \sqrt{225}}}}} = 4$ find the value of $x$

  1. 110

  2. 108

  3. 100

  4. 114
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have,

$\sqrt {10+\sqrt {25+\sqrt {x+\sqrt{154+\sqrt{225}}}}}=4$
$\Rightarrow \sqrt {10+\sqrt {25+\sqrt {x+\sqrt{154+15}}}}=4$
$\Rightarrow \sqrt {10+\sqrt {25+\sqrt {x+\sqrt{169}}}}=4$
$\Rightarrow \sqrt {10+\sqrt {25+\sqrt {x+13}}}=4$
On squaring both sides, we get
$10+\sqrt {25+\sqrt {x+13}}=16$
$\sqrt {25+\sqrt {x+13}}=6$
On squaring both sides, we get
$25+\sqrt {x+13}=36$
$\Rightarrow \sqrt {x+13}=11$
On squaring both sides, we get
$x+13=121$
$x=121-13=108$
Hence, $x=108$

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\sqrt[3]{5j - 7} = -\cfrac{1}{2}$, calculate the value of $j$.

  1. $1.375$
  2. $2.118$
  3. $2.599$
  4. $5.125$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $\sqrt [ 3 ]{ 5j-7 } =\dfrac { -1 }{ 2 } $

On cubing on both sides, we get
$5j-7=\dfrac { -1 }{ 8 } $
$\Rightarrow 5j=\dfrac { 55 }{ 8 } $
$\Rightarrow j=\dfrac { 11 }{ 8 } $
$\Rightarrow  j = 1.375$
Hence, option A is correct.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\sqrt[5]{\cfrac{g-1}{4}} = \cfrac{1}{3}$, then find the value of $g$.

  1. $0.984$
  2. $0.996$
  3. $1.004$
  4. $1.016$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $\sqrt [ 5 ]{ \dfrac { g-1 }{ 4 }  } =\dfrac { 1 }{ 3 } $

$ \Rightarrow  $ $\cfrac{g-1}{4} = \cfrac { 1 }{ { 3 }^{ 5 } } =\cfrac { 1 }{ 243 } $
$ \Rightarrow  $ $ g-1 = \cfrac{4}{243}$
$ \Rightarrow  $ $ g = 1 + \cfrac{4}{243} = 1.016$

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\cfrac{7}{m-\sqrt{3}} = \cfrac{\sqrt{3}}{m} + \cfrac{4}{2m}$, calculate the value of $m$.

  1. $-3.464$
  2. $-1.978$
  3. $-0.918$
  4. $1.978$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $\dfrac { 7 }{ m-\sqrt { 3 }  } =\dfrac { \sqrt { 3 }  }{ m } +\dfrac { 4 }{ 2m } =\dfrac { 4+2\sqrt { 3 }  }{ 2m } $
$\Rightarrow  14m=(4+2\sqrt { 3 } )m-\sqrt { 3 } (4+2\sqrt { 3 } )$
$ \Rightarrow (10-2\sqrt { 3 } )m=-4\sqrt { 3 } -6$
$\Rightarrow  m=\dfrac { -4\sqrt { 3 } -6 }{ 10-2\sqrt { 3 }  } =\dfrac { -12.928 }{ 6.536 } =-1.98$

Multiple choice maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

If $\left(\dfrac { 3 } { 4 }\right)^{th}$ of $x$ of $\left(\dfrac { 1 } { 4 }\right)^{th}$ of $35600 = 1668.75 ,$ find $x$

  1. $\dfrac { 2 } { 3 }$
  2. $\dfrac { 3 } { 4 }$
  3. $\dfrac { 2 } { 5 }$
  4. $\dfrac { 1 } { 4 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\dfrac { 3 }{ 4 } \times x\times \dfrac { 1 }{ 4 } \times 35600=1668.75$

$\Rightarrow x=\dfrac { 1668.75\times 16 }{ 3\times 35600 } =\dfrac { 1 }{ 4 } $      [D]

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

Compare the following pairs of surds. $\sqrt[8]{80}, \sqrt[4]{40}$    

  1. $\sqrt[8]{80} < \sqrt[4]{40}$
  2. $\sqrt[8]{80} \neq \sqrt[4]{40}$
  3. $\sqrt[8]{80} = \sqrt[4]{40}$
  4. $\sqrt[8]{80} > \sqrt[4]{40}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

   $\sqrt[8]{80}, \sqrt[4]{40}$
$={80}^{\frac{1}{8}}, {40}^{\frac{1}{4}}$
$={80}^{\frac{1}{8}}, {40}^{\frac{2}{8}}$
$={80}^{\frac{1}{8}}, {1600}^{\frac{1}{8}}$
Now,
   $80<1600$
$=>{80}^{\frac{1}{8}}<{1600}^{\frac{1}{8}}$
$=>\sqrt[8]{80}< \sqrt[4]{40}$

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

Compare the following pairs of surds $\sqrt[8]{12}, \sqrt[4]{6}$

  1. $\sqrt[8]{2} < \sqrt[4]{6}$
  2. $\sqrt[8]{8} < \sqrt[4]{6}$
  3. $\sqrt[8]{12} < \sqrt[4]{6}$
  4. $\sqrt[8]{12} < \sqrt[4]{4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To compare, convert to a common root: sqrt[8]{12} is 12^(1/8) and sqrt[4]{6} is 6^(1/4) = 6^(2/8) = 36^(1/8). Since 12 < 36, sqrt[8]{12} < sqrt[4]{6}.

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

Compare the following pair of surds:

$\sqrt[3]{6}, \sqrt[4]{8}$

  1. $\sqrt[3]{6} > \sqrt[4]{8}$
  2. $\sqrt[3]{6} > \sqrt[4]{4}$
  3. $\sqrt[3]{6} > \sqrt[3]{8}$
  4. $\sqrt[3]{4} > \sqrt[4]{8}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Convert to a common root (12th root): sqrt[3]{6} = 6^(4/12) = 1296^(1/12). sqrt[4]{8} = 8^(3/12) = 512^(1/12). Since 1296 > 512, the first is larger.