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 squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of which of the following numbers will be the least :

  1. $7\dfrac{58}{81}$
  2. $11\dfrac{14}{25}$
  3. $10\dfrac{1}{36}$
  4. $0.3481$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$A.$


$7\dfrac{58}{81}=\dfrac{625}{81}$


$\Rightarrow$  $\sqrt{\dfrac{625}{81}}=\dfrac{25}{9}=2.77$

$B.$

$11\dfrac{14}{25}=\dfrac{289}{25}$

$\Rightarrow$  $\sqrt{\dfrac{289}{25}}=\dfrac{17}{5}=3.4$

$C.$

$10\dfrac{1}{36}=\dfrac{361}{36}$

$\Rightarrow$  $\sqrt{\dfrac{361}{36}}=\dfrac{19}{6}=3.16$

$D.$

$0.3481=\dfrac{3481}{10000}$

$\Rightarrow$  $\sqrt{\dfrac{3481}{10000}}=\dfrac{59}{100}=0.59$

$\therefore$  We can see, $0.3481$  has least square root.

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

The positive square root of $( \sqrt { 48 } - \sqrt { 45 } )$ is _________.

  1. $\frac { \sqrt [ 4 ] { 3 } } { \sqrt { 2 } } ( \sqrt { 5 } - \sqrt { 3 } )$
  2. $\frac { \sqrt [ 4 ] { 3 } } { 2 } ( \sqrt { 5 } - \sqrt { 3 } )$
  3. $\frac { \sqrt { 2 } } { \sqrt [ 4 ] { 3 } } ( \sqrt { 5 } - \sqrt { 3 } )$
  4. $\frac { \sqrt [ 4 ] { 3 } } { \sqrt { 2 } } ( \sqrt { 5 } + \sqrt { 3 } )$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Solving $(\sqrt{48}-\sqrt{45})$

$4\sqrt{3}-3\sqrt{5}$

$\dfrac{\sqrt{3}}{2}(8-2\sqrt{15})$

$\dfrac{\sqrt{3}}{2}(3+5-2\sqrt{15})$

$\dfrac{\sqrt{3}}{2}(\sqrt{3^2}+\sqrt{5^2}-2\sqrt{5}\times\sqrt{3})$

$\dfrac{\sqrt3}{2}(\sqrt5-\sqrt3)^2$

$Now \ Finding\ Square \ Root$

$\pm{ \dfrac{3^{\frac{1}{4}}}{\sqrt2}(\sqrt5-\sqrt3)}$

$So \ it's \ positive \ root \ is \ $$ \dfrac{3^{\frac{1}{4}}}{\sqrt2}(\sqrt5-\sqrt3)$
Correct Answer is $A$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of 
$5-2\sqrt{6}$

  1. $\sqrt{13}-\sqrt{2}$
  2. $\sqrt{3}-\sqrt{2}$
  3. $\sqrt{5}-\sqrt{3}$
  4. $\sqrt{5}-\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$5-2\sqrt{6}=3+2-2\sqrt{6}$


$=({\sqrt{3}})^2+({\sqrt{2}})^2-2\sqrt{3}\times \sqrt {2}$

Using $(a-b)^2=a^2+b^2-2ab$


${5-2\sqrt{6}}=(\sqrt{3}-\sqrt{2})^2$

$\sqrt {5-2\sqrt{6}}=(\sqrt{3}-\sqrt{2})$

$So, \  the \  square \  root \  of \  (5-2\sqrt{6})=\sqrt{3}-\sqrt{2}$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

$\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}\, =\, ?$

  1. 0

  2. 1

  3. 2
  4. $2^{31/32}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\sqrt{2\, \times\, \sqrt{2\, \times\, \sqrt{2\, \times\, \sqrt{2\, \times\, 2^{1/2}}}}}$

$=\, \sqrt{2\, \times\, \sqrt{2\, \times\, \sqrt{(2\, \times\, 2^{3/4})}}}$
$=\, \sqrt{2\, \times\, \sqrt{2\, \times\, 2^{7/8}}}\, =\, \sqrt{2\, \times\, 2^{15/16}}\, =\, 2^{31/32}$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

By using the table for square root find the value of
$13.21$
$21.97$

  1. 3.63, 4.60

  2. 3.63, 4.69

  3. 3.53, 4.69

  4. 3.63, 4.19
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(i)

From square root table, Square root of 13.21 is:

 √13.21 = 3.6345

Therefore,

The square root of 13.21 is 3.63

(ii) From square root table, Square root of 21.97 is:

 √21.97 = 4.687

Therefore,

The square root of 21.97 is 4.69

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of $10$, correct to four places of decimal.

  1. 3.4623

  2. 3.1023

  3. 3.1693

  4. 3.1623
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$3.16227$
$3$$+3$ $10$$9$
$61$$+1$ $100$$61$
$626$$+6$ $3900$$3756$
$6322$$+2$ $14400$$12644$
$63242$$+2$-------------$632447$ $175600$$126484$---------------$4911600$$4427129$

$\sqrt{10}=3.16227\simeq 3.1623$
$\therefore$ The square root of $10$ correct to four places of decimal is $3.1623$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

The value of $\sqrt { \sqrt [ a ]{ { 4 }^{ { a }^{ { a }^{ 2 } } }\sqrt { { 6 }^{ { a }^{ 3 } }\sqrt [ { a }^{ 3 } ]{ { 12 }^{ { a }^{ 6 } }\sqrt [ { a }^{ 4 } ]{ { 18 }^{ { a }^{ 10 } } }  }  }  }  } $ is equal to

  1. $\sqrt {216}$
  2. $\sqrt {72}$
  3. $72$
  4. $216$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\begin{array}{l} =\sqrt { \sqrt [ a ]{ { { 4^{ { a^{ { a^{ 2 } } } } } }\sqrt { { 6^{ { a^{ 3 } } } }\sqrt [ { { a^{ 3 } } } ]{ { { { 12 }^{ { a^{ 6 } } } }\sqrt [ { { a^{ 4 } } } ]{ { { { 18 }^{ { a^{ 10 } } } } } }  } }  }  } }  }  \ =\sqrt { 4\times 6\times 18\times 12 }  \ =72 \ Hence, \ option\, \, C\, \, is\, correct\, \, answer. \end{array}$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of $\displaystyle 6 \frac{7}{8}$ correct to two decimal places 

  1. $2.62$
  2. $2.61$
  3. $2.63$
  4. $2.6$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

 $\displaystyle 6 \frac{7}{8}$
$=\dfrac{55}{8}$
$=6.875$

$2$$2$ $6.875$$4$
$46$  $6$ $287$$276$
$522$    $1150$$1044$
$6$                  

Therefore,
Square root of $\displaystyle 6 \frac{7}{8}=2.62$
                             

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

$\sqrt{(12\, +\, \sqrt{12\, +\, \sqrt{12\, +\, ........}})}\, =\, ?$

  1. 3

  2. 4

  3. 6

  4. Greater than 6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let $\sqrt{(12 + \sqrt{12 + \sqrt{12 + ........}})} = x$
Then, $\sqrt{12 + x} = x$
$ \Rightarrow 12 + x = x^2$
$\Rightarrow x^2 - x - 12 = 0$ 
$\Rightarrow (x - 4) (x + 3) = 0$
$\Rightarrow x = 4$       ...(neglecting $x = -3$)
Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of each of the following correct to three places of decimal.
$17$
$1.7$
$2.5$
$\displaystyle\frac{7}{8}$

  1. $4.153\;;\;1.304\;;\;1.581\;;\;0.935$
  2. $4.123\;;\;1.304\;;\;1.581\;;\;0.995$
  3. $4.123\;;\;1.304\;;\;1.581\;;\;0.935$
  4. $4.123\;;\;1.304\;;\;1.501\;;\;0.935$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$17$

$4.123$
$4$$+4$ $17$$16$
$822$$+2$ $100$$81$
$8243$ $1900$$1644$
$25600$$24729$                 $871$

$\therefore$ The square root of $17=4.123$

$1.7$

$0$ $1.303$
$1$$+1$ $1.7$$1$
$23$$+3$ $70$$69$
$2603$ $10000$$7809$
$1191$

$\therefore$ The square root of $1.7=1.303$

$2.5$

$0$ $1.581$
$1$$+1$ $2.5$$1$
$25$$+5$ $150$$125$
$308$$+8$ $2500$$2464$
$3161$ $3600$$3161$                 $439$

$\therefore$ The square root of $2.5=1.581$

$\frac{7}{8}$

$8$ $70$$64$ $0.875$
$60$$56$
$40$$40$             $0$
$0.935$
$0$ $0.875$$0$
$9$$9$ $87$$81$
$183$$+3$ $650$$549$
$1865$ $10100$$9325$
$925$

$\therefore$ The square root of $\frac{7}{8}=0.935$