Quantitative Aptitude · Mathematics

Algebraic Simplification

151 Questions

Algebraic simplification involves reducing mathematical expressions to their simplest forms using exponent rules and identities. Test takers must manipulate equations to find rapid solutions. This skill is frequently assessed in SSC, banking, and railway quantitative aptitude sections.

Exponent rulesSurd operationsAlgebraic identitiesTrigonometric simplificationLogical expressions

Algebraic Simplification Questions

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

Simplify the expression involving rational exponents:
${ \left( \displaystyle\frac { 25 }{ 64 }  \right)  }^{ { 1 }/{ 2 } }$

  1. $\displaystyle\frac { 25 }{ 8 } $

  2. Not a real number
  3. $\displaystyle\frac { 5 }{ 8 } $
  4. $\displaystyle\frac { 5 }{ 64 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\left(\dfrac{25}{64}\right)^\frac{1}{2}$ = $\left(\dfrac{5^2}{8^2}\right)^\frac{1}{2}$ = $\dfrac{5}{8}$

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Simplify: $( 16x ^{16} )^{\dfrac{3}{4}}$

  1. $8 x^{16}$
  2. $2 x^{12}$
  3. $8 x^{12}$
  4. $2 x^{16}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
${\left( 16{x}^{16} \right)}^{\cfrac{3}{4}} = {\left( {2}^{4} {x}^{16} \right)}^{\cfrac{3}{4}}$
$\Rightarrow \; = {\left( {2}^{4} \right)}^{\cfrac{3}{4}} {\left( {x}^{16} \right)}^{\cfrac{3}{4}}$

$\Rightarrow \; = {2}^\left( {4 \times \cfrac{3}{4}} \right)  {x}^\left({16 \times \cfrac{3}{4}} \right)$

$\Rightarrow \; = {2}^{3} \times {x}^{12}$

$\Rightarrow \; = 8{x}^{12}$
Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Simplify the following $(3r^2)\times (9r^2)^{3/2} \div (27r^{-3})^{1/3}$ and find the power of $r$.

  1. $5$
  2. $2$
  3. $7$
  4. $6$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We have,
$(3r^2)\times (9r^2)^{3/2}\div(27r^{-3})^{1/3}\$

$\Rightarrow (3r^2)\times ((3r)^2)^{3/2}\div(3^3r^{-3})^{1/3}\$
$\Rightarrow (3r^2)\times (3r)^{3}\div(3r^{-1})\$

$\Rightarrow (3r^2)\times (3^3r^3)\times(3^{-1}r)\\$
$\Rightarrow 27r^6$

So, the power of $r$ is $6$.

Hence, this is the answer.

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Simplify: $\displaystyle \left ( -a \right )^{9}\times \left ( -b \right )^{9}$ 

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

$(-a)^9 \times (-b)^9 = [ -a \times -b]^9$


= $ [ a \times b]^9$

=$ (ab)^9$
So, option $A$ is correct.

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Simplify and give reasons:
${ \left( \cfrac { 1 }{ 2 }  \right)  }^{ -3 }\times { \left( \cfrac { 1 }{ 4 }  \right)  }^{ -3 }\times { \left( \cfrac { 1 }{ 5 }  \right)  }^{ -3 }\quad $

  1. ${40}^{3}$
  2. ${40}^{-3}$
  3. ${40}^{6}$

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

we know,

$(\dfrac{a}{b})^{-m}=(\dfrac{b}{a})^{m}$
So,
$(\dfrac{1}{2})^{-3}=(\dfrac{2}{1})^{3}=2^{3}$

$(\dfrac{4}{1})^{-3}=(\dfrac{4}{1})^{3}=4^{3}$

$(\dfrac{5}{1})^{-3}=(\dfrac{5}{1})^{3}=5^{3}$
now we know,

\$a^{m}b^{m}*c^{m}=(abc)^{m}$

$\implies(2)^{3}(4)^{3}*(5)^{3}$

$=(40)^{3}$

Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

If $\displaystyle x=\frac{4\sqrt{2}}{\sqrt{2}+1}$ then find the value of $\displaystyle \frac{1}{\sqrt{2}}\left ( \frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x-2\sqrt{2}} \right )$

  1. $\displaystyle \sqrt{2}$
  2. $12+8\displaystyle \sqrt{2}/5$
  3. $12-8\displaystyle \sqrt{2}$
  4. $\displaystyle \frac{16\sqrt{2}+24}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x=\frac{4\sqrt{2}}{\sqrt{2+1}}$
$\frac{1}{\sqrt{2}}\left ( \frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x-2\sqrt{2}} \right )$
Put the value of x
$\frac{1}{\sqrt{2}}\left ( \frac{\frac{4\sqrt{2}}{\sqrt{2+1}}+2}{\frac{4\sqrt{2}}{\sqrt{2+1}}-2}+\frac{\frac{4\sqrt{2}}{\sqrt{2+1}}+2\sqrt{2}}{\frac{4\sqrt{2}}{\sqrt{2+1}}-2\sqrt{2}} \right )$
=$\frac{1}{\sqrt{2}}\left ( \frac{4\sqrt{2}+2\sqrt{2}+2}{4\sqrt{2}-2\sqrt{2}-2} \right )+\left ( \frac{4\sqrt{2}+4+2\sqrt{2}}{4\sqrt{2}-4-2\sqrt{2}} \right )$
=$\frac{6\sqrt{2}+2}{2\sqrt{2-2}}+\frac{6\sqrt{2}+4}{2\sqrt{2}-4}$
=$\frac{1}{\sqrt{2}}\left ( \frac{32-24\sqrt{2}}{16-12\sqrt{2}} \right )$
=$\sqrt{2}$

Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

If a=2, b=3, c=4, then the difference between $\displaystyle 2\frac{3}{4}$ and $\displaystyle b\frac{a}{c}$ is

  1. $-1/4$
  2. $1/4$
  3. $-3/4$
  4. $5/4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If a=2 ,b=3 and c=4

Then $b\tfrac{a}{c}=3\tfrac{2}{4}=3\tfrac{1}{2}$
Then difference between=$2\tfrac{3}{4}-3\tfrac{1}{2}=\frac{11}{4}-\frac{7}{2}=\frac{11-14}{4}=\frac{-3}{4}$

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

Simplify: $\dfrac{{4 + \sqrt 5 }}{{4 - \sqrt 5 }} + \dfrac{{4 - \sqrt 5 }}{{4 + \sqrt 5 }}$

  1. $\dfrac {42}{11}$
  2. $\dfrac {40}{11}$
  3. $\dfrac {39}{25}$
  4. $\dfrac {16}{25}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{4+\sqrt5}{4-\sqrt5} = \dfrac{(4+\sqrt5)}{(4-\sqrt5)} \dfrac{(4+\sqrt5)}{(4+\sqrt5)}$    ...... rationalizing numerator and the denominator


$= \dfrac{(4+\sqrt5)^2}{16-5} = \dfrac{(4+\sqrt5)^2}{11} $

$\dfrac{4-\sqrt5}{4+\sqrt5} = \dfrac{(4-\sqrt5)}{(4+\sqrt5)} \dfrac{(4-\sqrt5)}{(4-\sqrt5)}$    ...... rationalizing numerator and the denominator, 

$= \dfrac{(4-\sqrt5)^2}{16-5} = \dfrac{(4-\sqrt5)^2}{11} $


$\dfrac{4+\sqrt5}{4-\sqrt5} +\dfrac{4-\sqrt5}{4+\sqrt5} = \dfrac{(4+\sqrt5)^2 +(4-\sqrt5)^2}{11} =\dfrac{16+5+16+5}{11} = \dfrac{42}{11}$

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

What is the value of $\dfrac {1}{1 + \sqrt {2} + \sqrt {3}} + \dfrac {1}{1 - \sqrt {2} + \sqrt {3}}$?

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

The value of $\dfrac {1}{(1 + \sqrt {3})+\sqrt {2}} + \dfrac {1}{(1 + \sqrt {3}) - \sqrt {2}}$ is
$=\dfrac {(1 + \sqrt {3} - \sqrt {2}) + (1 + \sqrt {3} + \sqrt {2})}{(1 + \sqrt {3})^{2} - (\sqrt {2})^{2}}$
$= \dfrac {2(1 + \sqrt {3})}{1 + 3 + 2\sqrt {3} - 2}$
$= \dfrac {2(1 + \sqrt {3})}{2(1 + \sqrt {3})} = 1$

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

Simplify: $\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} -\dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}}$

  1. $0$
  2. $1$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} -\dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}}\=\dfrac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{(\sqrt{10} + \sqrt{3})(\sqrt{10} - \sqrt{3})} - \dfrac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{(\sqrt{6} + \sqrt{5})(\sqrt{6} - \sqrt{5})} -\dfrac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{(\sqrt{15} - 3\sqrt{2})(\sqrt{15} + 3\sqrt{2})}\=\dfrac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{10-3} - \dfrac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{6-5} -\dfrac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{15-18}\=\dfrac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{7} - \dfrac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{1} +\dfrac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{3}$

$=\dfrac{21\sqrt{30}-63-425\sqrt{30}+210+21\sqrt{30}-18*7}{21}\=\dfrac{21}{21}=1$

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

Simplify:
$\dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

  1. $4\sqrt{6}$
  2. $10$
  3. $2$
  4. $\dfrac{4\sqrt{6}}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}=\dfrac{(\sqrt{3} + \sqrt{2})^2+(\sqrt{3} - \sqrt{2})^2}{3-2}=\dfrac{3+2+3+2}{1}=10$