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 complex numbers and linear inequations identities of complex numbers powers of imaginary unit i algebra of complex numbers

Simplify the following :


$\left(\dfrac{1 \, + \, i}{1 \, - \, i}\right)^{4n \, + \, 1}$

  1. $1$
  2. $i$
  3. $0$

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

For $n$ is positive integer, $4n+1$ is an odd integer$(5,9,13,....)$


Now,


$\left(\dfrac{1+i}{1-i}\right)^{4n+1}$

$\implies \left(\dfrac{(1+i)(1+i)}{(1-i)(1+i)}\right)^{4n+1}$

$\implies \left(\dfrac{(1-1+2i}{1-(i^2)}\right)^{4n+1}$

$\implies \left(\dfrac{2i}{2}\right)^{4n+1}\implies (i)^{4n+1}=i$.................[putting $n=4,9,13.....$]

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

On simplification the product of given expression $\left (x - \dfrac {1}{x}\right )\left (x + \dfrac {1}{x}\right )\left (x^{2} + \dfrac {1}{x^{2}}\right )$ is ________.

  1. $x^{3} - \dfrac {1}{x^{3}}$
  2. $x^{3} + \dfrac {1}{x^{3}}$
  3. $x^{4} - \dfrac {1}{x^{4}}$
  4. $x^{4} + \dfrac {1}{x^{4}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
We need to find value of $\left (x-\dfrac{1}{x}\right)\left (x+\dfrac{1}{x}\right)\left (x^{2}+\dfrac{1}{x^{2}}\right)$
We know the identity $a^{2}-b^{2} = (a-b)(a+b)$

Then applying this to first two terms:

$\left (x^{2}-\dfrac{1}{x^{2}}\right)\left (x^{2}+\dfrac{1}{x^{2}}\right)$

Again applying same identity:

$x^{4}-\dfrac{1}{x^{4}}$

Hence, option C is correct.

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Simplify : $\frac{(-18\frac{1}{3}\times 2\frac{8}{11})}{|\frac{3}{5}+(\frac{-9}{10})| + |-(\frac{-3}{5})|}$

  1. 63$\frac{4}{81}$
  2. -23$\frac{7}{9}$
  3. -67$\frac{7}{9}$
  4. 12$\frac{6}{17}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\cfrac {\left(-18\cfrac{1}{3}\times 2\cfrac {8}{11}\right)}{\left|\cfrac {3}{5}+\left(\cfrac {-9}{10}\right)\right|+\left|-\left(\cfrac{-3}{5}\right)\right|}$

$=\cfrac{-\cfrac{55}{3}\times \cfrac{30}{11}}{\left|\cfrac {6-9}{10}\right|+\cfrac {3}{5}}$
$=\cfrac {\cfrac {-55\times 30}{33}}{\cfrac {3}{10}+\cfrac {3}{5}}$
$=\cfrac {\cfrac {-55\times 30}{33}}{\cfrac{9}{50}}$
$=\cfrac {-55\times 30\times 50}{33\times 9}$
$=\cfrac {-5\times 10\times 50}{9}$
$=-67\cfrac {7}{9}$

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

The simplified form of the expression $\sqrt { \sqrt [ 3 ]{ 729{ x }^{ 12 } }  } -\dfrac { { x }^{ -2 }-{ x }^{ -3 } }{ { x }^{ -4 }-{ x }^{ -5 } } $ is

  1. ${ 3x }^{ 2 }$
  2. ${ 3x }^{ 3 }$
  3. ${ 2x }^{ 2 }$
  4. ${ 4x }^{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sqrt{(729x^{12})^{\frac{1}{3}}}-\cfrac{x^{-2}-x^{-3}}{x^{-4}-x^{-5}}$


$=\sqrt{(3^6x^{12})^{\frac{1}{3}}}-\cfrac{x^{-2}-x^{-3}}{x^{-4}-x^{-5}}$


$=\sqrt{(3^2x^4)}-\cfrac{x^{-2}(1-x^{-1})}{x^{-4}(1-x^{-1})}$

$=3x^2-x^2$

$\=2x^2$

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

Assuming  that x,y,z  are positive real numbers,simplify the following :


$ (\sqrt{x})^{-2/3}\sqrt{y^{4}}\div \sqrt{xy^{-1/2}} $

  1. $ \dfrac{y^{9/4}}{x^{5}} $
  2. $ \dfrac{y^{9/4}}{x^{5/6}} $
  3. $ \dfrac{y^{9/4}}{x^{-5/6}} $
  4. $ \dfrac{y^{-9/4}}{x^{5/6}} $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

$(\sqrt{x})^{-\frac{2}{3}}\sqrt{y^4}\div \sqrt{xy^{-\frac{1}{2}}}$

$=\dfrac{x^{-\frac{1}{3}}y^2}{x^{\frac{1}{2}}y^{-\frac{1}{4}}}$

$=x^{-\frac{1}{3}-\frac{1}{2}}y^{2+\frac{1}{4}}$

$=x^{-\frac{5}{6}}y^{\frac{9}{4}}$

$=\dfrac{y^{\frac{9}{4}}}{x^{\frac{5}{6}}}$
Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

Simplify : $[ 0.9 - { 2.3 - 3.2 - 3.2 - ( 7.1- 5.4 - 3.5 ) } ]$

  1. 0.18

  2. 1.8

  3. 0

  4. 2.6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle \left [ 0.9-\left { 2.3-3.2-\left ( 7.1-5.4-3.5 \right ) \right } \right ]$


=$\displaystyle \left [ 0.9-\left { 2.3-3.2-\left ( 7.1-8.9 \right ) \right } \right ]$

= $\displaystyle \left [ 0.9-\left { 2.3-3.2-\left ( -1.8 \right ) \right } \right ]$

= $\displaystyle \left [ 0.9-\left { 2.3-3.2+1.8 \right } \right ]$

= $\displaystyle \left [ 0.9-\left { 4.1-3.2 \right } \right ]$

= $\displaystyle \left [ 0.9-0.9 \right ]=0 $

Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

Simplify the following :

$0.4 \times \displaystyle \frac{7}{3} \div \frac{15}{8}  $ of $  \left ( \dfrac{7}{5} - \dfrac{4}{3} \right )$.

  1. $\displaystyle 5 \frac{8}{3375}$
  2. $\displaystyle 7 \frac{8}{15}$
  3. $\displaystyle 7 \frac{7}{15}$
  4. $\displaystyle 5 \frac{7}{3375}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The given expression is

$0.4\times \dfrac { 7 }{ 3 } \div \dfrac { 15 }{ 8 } $ of $\left( \dfrac { 7 }{ 5 } -\dfrac { 4 }{ 3 }  \right) $

$=\dfrac { 4 }{ 10 } \times \dfrac { 7 }{ 3 } \div \dfrac { 15 }{ 8 } $ of $\dfrac { 1 }{ 15 } $

$ =\dfrac { 2 }{ 5 } \times \dfrac { 7 }{ 3 } \div \dfrac { 1 }{ 8 } $

$ =\dfrac { 2 }{ 5 } \times \dfrac { 7 }{ 3 } \times \dfrac { 8 }{ 1 } $

$ =\dfrac { 112 }{ 15 } $

$=7\dfrac { 7 }{ 15 } $

Multiple choice maths the integers properties of integer multiplication properties of addition and subtraction of integers propertie of integers

Simplify using associative property:
$(-5) \times [(-8) \times 2] =$  $....$

  1. $40$
  2. $-80$
  3. $80$
  4. $-40$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(-5) \times [(-8)\times 2] = (-5)\times (2\times (-8))$ (commutative property)
$= [(-5)\times 2] \times (-8)$ (associative property)
$= (-10) \times (-8)$
$= 80$
(Here, we used various properties to skip $(-5)\times (-8)$ which could have been time consuming).

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

Simplify the expression $2\dfrac{1}{4}\times \dfrac{5}{12}+\dfrac{1}{2}$

  1. $\dfrac{23}{16}$
  2. $5\dfrac{5}{2}$
  3. $4\dfrac{3}{3}$
  4. $3\dfrac{1}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$2\cfrac { 1 }{ 4 } \times \cfrac { 5 }{ 12 } +\cfrac { 1 }{ 2 } $


$=\cfrac { 9 }{ 4 } \times \cfrac { 5 }{ 12 } +\cfrac { 1 }{ 2 } $

$ =\cfrac { 15 }{ 16 } +\cfrac { 1 }{ 2 } $

$ =\cfrac { 15+8 }{ 16 } $

$ =\cfrac { 23 }{ 16 } $

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

Multiply and reduce to lowest form:
$\cfrac { 2 }{ 3 } \times 5\cfrac { 1 }{ 5 } $

  1. $3\cfrac { 7 }{ 15 } $
  2. $7\cfrac { 3 }{ 15 } $
  3. $\cfrac { 7 }{ 15 } $
  4. $3\cfrac { 3 }{ 15 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{2}{3}\times 5\dfrac{1}{5}$


Can be written as


$\dfrac{2}{3}\times \dfrac{26}{5}$

$=\dfrac{52}{15}$

$=3\dfrac{7}{15}$

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

$4\frac{4}{5}\div\frac{3}{5}$ of $5+\frac{4}{5}\times\frac{3}{10} -\frac{1}{5}$ is simplified, then the result is

  1. $1\frac{16}{25}$
  2. $1\frac{17}{25}$
  3. $\frac{40}{25}$
  4. $\frac{42}{25}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Apply BODMAS
$4\frac{4}{5}\div\frac{3}{5}$ of $5+\frac{4}{5}\times\frac{3}{10} -\frac{1}{5}$
$=\frac{24}{5}\div3+\frac{4}{5\times\frac{3}{10}-\frac{1}{5}}$
$=\frac{8}{5}+\frac{6}{25}-\frac{1}{5}$
$=\frac{41}{25}$
$=1\frac{16}{25}$
Option 'A' is the answer