Quantitative Aptitude · Mathematics

Algebraic Simplification

146 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 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 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

Simplify: $\dfrac{5}{11} + 4\dfrac{3}{9} $

  1. $\dfrac{158}{33}$
  2. $\dfrac{168}{33}$
  3. $\dfrac{178}{33}$
  4. $\dfrac{148}{33}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\displaystyle \frac{5}{11}+4\frac{3}{9}=\frac{5}{11}+\frac{36+3}{9}=\frac{5}{11}+\frac{39}{9}$

$\displaystyle =\frac{5\times 9+11\times 39}{9\times 11}=\frac{45+429}{99}=\frac{474}{99}=\frac{158}{33}$

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

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

Multiple choice

Simplify the expression: (2x + 3y - 4x + 5y)

  1. \(-2x + 8y\)
  2. \(-2x + 2y\)
  3. \(8x - 2y\)
  4. \(8x + 2y\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Combine like terms to simplify the expression: (2x + 3y - 4x + 5y = -2x + 8y).