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 parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

The value of $\left (-\dfrac {7}{2}\right )^{-1}$ is _________.

  1. $-1$
  2. $\dfrac {7}{2}$
  3. $-\dfrac {2}{7}$
  4. $\dfrac {-7}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find $\left(-\dfrac{7}{2}\right)^{-1}$


Any fraction raised to negative power yields same result as of its reciprocal with modulus of the power.

$\therefore \left(-\dfrac{7}{2}\right)^{-1} = -\dfrac{2}{7}$

Multiple choice maths parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

The value of the expression $\sqrt {34-24\sqrt 2}\times (4+3\sqrt 2)$ is

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

$\sqrt {34-24\sqrt 2}\times (4+3\sqrt 2)$


$=\sqrt {34-24\sqrt 2}\sqrt {(4+3\sqrt 2)^2}$

$=\sqrt {(34-24\sqrt 2)(16+18+24\sqrt 2)}$

$=\sqrt {(34-24\sqrt 2)(34+24\sqrt 2)}$

$=\sqrt {(34)^2(24\sqrt 2)^2}$

$=\sqrt {1156-1152}=\sqrt 4=2$

Multiple choice maths parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

When simplified, the product $\left( 1-\cfrac { 1 }{ 3 }  \right) \left( 1-\cfrac { 1 }{ 4 }  \right) \left( 1-\cfrac { 1 }{ 5 }  \right) ...\left( 1-\dfrac 1n \right) $ becomes

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

$\left( 1-\cfrac { 1 }{ 3 }  \right) \left( 1-\cfrac { 1 }{ 4 }  \right) \left( 1-\cfrac { 1 }{ 5 }  \right) ...\left( 1- \cfrac 1n \right) =\cfrac { 2 }{ 3 } .\cfrac { 3 }{ 4 } .\cfrac { 4 }{ 5 } ....\cfrac { n-2 }{ n-1 } .\cfrac { n-1 }{ n } =\cfrac { 2 }{ n } $

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Read out each of the following numbers carefully and specify the natural numbers in it.
$87, 54, 0, -13, -4.7, \sqrt{7}, 2{1}{7}, \sqrt{15}, -{8}{7}, 3\sqrt{7}, 4.807, 0.002, \sqrt{16}$ and $2+\sqrt{3}.$

  1. $0,87,54,\sqrt{16}$
  2. $87, 54,$ $\sqrt{16}$, $217$
  3. $0, -13, -4,7, 217, 54, 87$
  4. $\sqrt{7}$, $\sqrt{15}$, $3 \sqrt{7}$, $\sqrt{16}$, $2 + \sqrt{3}$,
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Natural numbers from the given list are 87, 54,  $\sqrt { 16 } =4$ and 217

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Which of the following number is different from others?

  1. $\sqrt 7$
  2. $\sqrt 6$
  3. $\sqrt {25}$
  4. $\sqrt{10}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sqrt{7}$ is an irrational number

$\sqrt{6}$ is an irrational number
$\sqrt{10}$ is an irrational number
$\sqrt{25}=5$ is different from others because others are irrational number but $\sqrt{25}$ is a rational number
Hence, option C is correct.

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

$\left ( 2+\sqrt{5} \right )\left ( 2+\sqrt{5} \right )$ expression is :

  1. A rational number

  2. A whole number

  3. An irrational number

  4. A natural number
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ (2+\sqrt { 5 } ) }^{ 2 }\ =4+5+4\sqrt { 5 } \ =9+4\sqrt { 5 } $

In the above equation $4\sqrt { 5 } $ is irrational number so $9+4\sqrt { 5 } $ will also be irrational number 
So correct answer is option C.

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

$\sqrt{21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}}=$...........

  1. $\sqrt{5}-2+2\sqrt{3}$
  2. $\sqrt{5}-\sqrt{4}-\sqrt{12}$
  3. $-\sqrt{5}+\sqrt{4}+\sqrt{12}$
  4. $-\sqrt{5}-\sqrt{4}+\sqrt{12}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The expression is sqrt(21 - 4*sqrt(5) + 8*sqrt(3) - 4*sqrt(15)). This is of the form sqrt((a+b+c)^2) = |a+b+c|. Expanding (sqrt(5) - 2 - 2*sqrt(3))^2 gives 5 + 4 + 12 - 4*sqrt(5) - 4*sqrt(15) + 8*sqrt(3) = 21 - 4*sqrt(5) + 8*sqrt(3) - 4*sqrt(15). Thus the square root is |sqrt(5) - 2 - 2*sqrt(3)|, which equals -sqrt(5) + 2 + 2*sqrt(3).

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$
$\Rightarrow  m=\dfrac { -4\sqrt { 3 } -6 }{ 10-2\sqrt { 3 }  } =\dfrac { -12.928 }{ 6.536 } =-1.98$