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 square root square root of perfect square finding square root of a number square root of a perfect square

The value  of  $\sqrt {11 - \sqrt{112} }=  $

  1. $2 + \sqrt{7}$
  2. $2 - \sqrt{7}$
  3. $ \sqrt{7} - 2$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$11-\sqrt {112}$

$=11-\sqrt {4\times 28}$

$=11-\sqrt {4}\times \sqrt {28}$

$=11-2\times \sqrt {28}$

$=11-2\times \sqrt {7}\times \sqrt {4}$

$=7+4-2\times \sqrt {7}\times \sqrt {4}$

$=(\sqrt {7})^2 +(\sqrt {4})^2 -2\times \sqrt {7}\times \sqrt {4}$

$=(\sqrt {7}-\sqrt {4})^2$

Thus, $11-\surd {112}=(\surd {7} -\surd {4})^2$

Hence,

$\sqrt {11-\sqrt {112}}=\sqrt {(\sqrt {7}-\sqrt {4})^2}=\sqrt {7}-\sqrt {4}=\sqrt {7}-2$
Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

If ${\left( {\dfrac{m}{n}} \right)^{\dfrac{3}{8}}} + {\left( {\dfrac{n}{m}} \right)^{\dfrac{3}{8}}} = 9$ then find the value of ${\left( {\dfrac{m}{n}} \right)^{\dfrac{3}{4}}} + {\left( {\dfrac{n}{m}} \right)^{\dfrac{3}{4}}}$

  1. $79$
  2. $72$
  3. $83$
  4. $84$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$(\dfrac{m}{n})^\dfrac{3}{8}+(\dfrac{n}{m})^\dfrac{3}{8}=9$
$[(\dfrac{m}{n})^\dfrac{3}{8}+(\dfrac{n}{m})^\dfrac{3}{8}]^{2} =9^{2}$
$ ((\dfrac{m}{n})^\dfrac{3}{8})^{2}+((\dfrac{n}{m})^\dfrac{3}{8})^{2}+2((\dfrac{m}{n})^\dfrac{3}{8})((\dfrac{n}{m})^\dfrac{3}{8})=81$
$ (\dfrac{m}{n})^\dfrac{3}{4}+(\dfrac{n}{m})^\dfrac{3}{4}=81-2=79$
Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

The square root of sum of the digits in the square of $121$ is

  1. $4$
  2. $3$
  3. $6$
  4. $9$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
${ \left( 121 \right)  }^{ 2 }$

$={ \left( 100+21 \right)  }^{ 2 }$     

$={ 100 }^{ 2 }+{ 21 }^{ 2 }+2\left( 100 \right) \left( 21 \right) $      $[\because (a+b)^2= a^2+2ab+b^2]$

$=14641$

Sum of digits $=1+4+6+4+1=16$

Square root$=\sqrt { 16 } =4$
Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Find the square root of $225$ using "Repeated Subtraction".

  1. $11$
  2. $15$
  3. $5$
  4. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\\225-1=224\\224-3=221\\221-5=216\\216-7=209\\209-9=200\\200-11=189\\189-13=176\\176-15=161\\161-17=144\\144-19=125\\125-21=104\\104-23=81\\81-25=56\\56-27=29\\29-29=0\\\>Total\>steps\>of\>=15\>\\hence\>\sqrt{225}=15$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

The square root of $42\, \displaystyle \frac{583}{1369}$ is :

  1. $6\, \displaystyle \frac{19}{37}$
  2. $4\, \displaystyle \frac{2}{11}$
  3. $7\, \displaystyle \frac{2}{121}$

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

$\sqrt{42\, \displaystyle \cfrac{583}{1369}}\, =\, \sqrt{\displaystyle \cfrac{58081}{1369}}$
$=\, \displaystyle \cfrac{\sqrt{58081}}{\sqrt{1369}}$
$=\, \displaystyle \cfrac{241}{37}\, =\, 6\, \displaystyle \cfrac{19}{37}$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

If $\sqrt{49}=7$, then find the value of $\sqrt{49}+\sqrt{0.49}+\sqrt{0.0049}+\sqrt{0.000049}$

  1. 7777

  2. 77.77

  3. 777.7

  4. 7.777
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\sqrt{49}+\sqrt{0.49}+\sqrt{0.0049}+\sqrt{0.000049}\=7+\sqrt { \displaystyle\frac { 49 }{ 100 }  } +\sqrt {\displaystyle \frac { 49 }{ 10000 }  } +\sqrt { \displaystyle\frac { 49 }{ 1000000 }  } \ =7+\displaystyle\frac { \sqrt { 49 }  }{ \sqrt { 100 }  } +\displaystyle\frac { \sqrt { 49 }  }{ \sqrt { 10000 }  } +\displaystyle\frac { \sqrt { 49 }  }{ \sqrt { 1000000 }  } \ =7+\displaystyle\frac { 7 }{ 10 } +\displaystyle\frac { 7 }{ 100 } +\displaystyle\frac { 7 }{ 1000 } \ =7+0.7+0.07+0.007\ =7.777$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Find the square roots of $100\;and\;169$ by method of repeated substraction.

  1. $10; 13$
  2. $10; 17$
  3. $20; 13$
  4. $20; 17$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For $\sqrt{100}$
$100-1=99$
$99-3=96$
$96-5=91$
$91-7=84$
$84-9=75$
$75-11=64$
$64-13=51$
$51-15=36$
$36-17=29$
$19-19=0$

For $\sqrt{169}$
$169-1=168$
$168-3=165$
$165-5=160$
$160-7=153$
$153-9=144$
$144-11=133$
$133-13=120$
$120-15=105$
$105-17=88$
$88-19=69$
$69-21=48$
$48-23=25$
$25-25=0$
From $100\;and\;169$ we have substracted successive odd numbers starting from $1$ and obtained $0$ at $10th\;and\;13th$ steps, respectively.
So, $\sqrt{100}=10\;and\;\sqrt{169}=13$. 

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Find the square root of $100$ by the method of repeated substraction.

  1. $10$.
  2. $11$
  3. $9$
  4. $12$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know that the sum of the first n odd natural numbers is $n^{2}$
From $100$, we subtract successive odd numbers starting from $1$ as under
$100-1=99\;\;\;99-3=96$
$96-5=91\;\;\;\;91-7=84$
$84-9=75\;\;\;\;75-11=64$
$64-13=51\;\;\;51-15=36$
$36-17=19\;\;\;19-19=0$
and obtain $0$ at $10th$ step.
$\therefore\;\sqrt{100}=10$.
From $169$, we subtract successive odd numbers starting from $1$ as under
$169-1=168\;\;\;168-3=165$
$165-5=160\;\;\;\;160-7=153$
$153-9=144\;\;\;\;144-11=133$
$133-13=120\;\;\;120-15=105$
$105-17=88\;\;\;88-19=69$
$69-21=48\;\;\;48-23=25$
$25-25=0$
and obtain $0$ at $13th$ step.
$\therefore\;\sqrt{169}=13$.





Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Complete the repeated subtraction to find the square root of $225$.

  1. $9$
  2. $45$
  3. $25$
  4. $15$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$225 -1 = 224$ ,

$224 -3 = 221 $,
$221-  5 = 216$, 
$216 -7 = 209$,
$209 - 9= 200$ ,
$200 -11 = 189$
$189-  13 = 176$ , 
$176-  15 = 161$, 
$161 -17 = 144$,
$144- 19 = 125$
$125 - 21 = 104 $
$ 104-  23 = 81 $
$81-  25 = 56$
$ 56-  27 = 29$, 
$29 -29 = 0 = 15$ 
Therefore, D is the correct answer.