Quantitative Aptitude

Number System and Simplification

585 Questions

Number system and simplification questions assess mathematical proficiency in handling fractions, decimals, and complex algebraic expressions. Test takers must simplify surds, calculate reciprocals, and solve intricate number pattern equations. This foundational quantitative aptitude topic is critical for achieving high scores in SSC and banking exams.

Fraction simplificationDecimal operationsSurds and indicesPercentage calculationsComplex number algebra

Number System and Simplification Questions

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

Find the value of: $\displaystyle \left [ \left ( -1 \right )^{2}\times \left ( -1 \right )^{3}\times \left ( -1 \right )^{4} \right ]^{6}$

  1. $3$
  2. $-1$
  3. $1$
  4. $\displaystyle 3^{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle \left [ \left ( -1 \right )^{2}\times \left ( -1 \right )^{3}\times \left ( -1 \right )^{4} \right ]^{6}$

=$\displaystyle \left [ \left ( -1 \right )^{12}\times \left ( -1 \right )^{18}\times \left ( -1 \right )^{24} \right ]$
=$\displaystyle \left [ 1\times 1\times 1 \right ]$ =$1$
So, option $C$ is correct.

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

The value of $\displaystyle \left (-4  \right ) ^{3}\times \left ( -3 \right )^{3}$ is _____?

  1. $\displaystyle 12^{3}$
  2. $\displaystyle -12^{3}$
  3. $\displaystyle -7^{3}$
  4. $\displaystyle 7^{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle \left ( -4 \right )^{3}\times \left ( -3 \right )^{3}=\left ( -4\times -3 \right )^{3}$

$=\displaystyle \left ( -4\times -3 \right )^{3}$
$=\displaystyle 12^3$

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

The value of $\displaystyle \left [ \left ( \frac{-2}{5} \right )^{3} \right ]^{2}$ is:

  1. $\displaystyle -\frac{2}{5}$
  2. $\displaystyle -\frac{32}{3125}$
  3. $\displaystyle\frac{64}{15625}$
  4. $\displaystyle -\frac{64}{15625}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle \left [ \left ( \frac{-2}{5} \right )^{3} \right ]^{2}=\left ( \frac{-2}{5} \right )^{6}=\frac{\left ( -2 \right )^{6}}{5^{6}}=\frac{64}{15625}$

Hence, option $C$ is correct.

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

Evaluate $\displaystyle\left [ \left ( \frac{-3}{7} \right )^{-1} \right ]^{2}$

  1. $\displaystyle\left ( \frac{-9}{49} \right )$
  2. $\displaystyle\left ( \frac{9}{49} \right )$
  3. $\displaystyle\left ( \frac{-49}{9} \right )$
  4. $\displaystyle\left ( \frac{49}{9} \right )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle\left [ \frac{-3}{7} ^{-1}\right ]^{2}=\left [ \left ( \frac{-7}{3} \right )^{1} \right ]^{2}=\left ( \frac{-7}{3} \right )^{2}=\frac{49}{9}$

Hence, option $D$ is correct.

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

The value of $\displaystyle \left ( \frac{2}{3}\right )^{-5}$ is:

  1. $\displaystyle -\frac{32}{243}$
  2. $\displaystyle \frac{32}{243}$
  3. $\displaystyle -\frac{243}{32}$
  4. $\displaystyle \frac{243}{32}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle\left ( \frac{2}{3} \right )^{-5}=\left ( \frac{3}{2} \right )^{5}=\frac{3^{5}}{2^{5}}=\frac{243}{32}$

Hence, option $D$ is correct.

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

$\displaystyle \left ( 16\div 15 \right )^{3}$ can also be expressed as:

  1. $\displaystyle 16^{3}\div 15^{3} $
  2. $\displaystyle 16^{3}\div 15 $
  3. $\displaystyle 16\div 15^{3} $
  4. $\displaystyle 15^{3}\div 16^{3} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle\left ( 16\div 15 \right )^{3}=16^{3}\div 15^{3}$ 


This is quotient law of exponents.
Hence, option $A$ is correct

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

$\dfrac{1}{1+x^{(b-a)}+x^{(c-a)}}+\dfrac{1}{1+x^{(a-b)}+x^{(c-b)}}+\dfrac{1}{1+x^{(b-c)}+x^{(a-c)}} = ?$

  1. $0$
  2. $1$
  3. $x^{a-b-c}$

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

Given Exp. = $\dfrac{1}{\begin{pmatrix}1+\dfrac{x^b}{x^a}+\dfrac{x^c}{x^a}\end{pmatrix}} + \dfrac{1}{\begin{pmatrix}1+\dfrac{x^a}{x^b}+\dfrac{x^c}{x^b}\end{pmatrix}} + \dfrac{1}{\begin{pmatrix}1+\dfrac{x^b}{x^c}+\dfrac{x^a}{x^c}\end{pmatrix}}$
$= \dfrac{x^a}{(x^a+x^b+x^c)}+\dfrac{x^b}{(x^a+x^b+x^c)} + \dfrac{x^c}{(x^a+x^b+x^c)}$
$= \dfrac{x^a+x^b+x^c}{(x^a+x^b+x^c)}$
$= 1.$

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

Find:$\dfrac{\sqrt[3]{108}\times \sqrt[6]{4}}{\sqrt[4]{81}}$

  1. $ 2$
  2. $\frac{\sqrt[6]{4}}{\sqrt[2]{3}}$
  3. $ \sqrt[4]{6}$
  4. $\frac{\sqrt[4]{2}}{\sqrt[2]{3}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$(\frac{\sqrt[3]{180}\times \sqrt[6]{4}}{\sqrt[4]{81}})\\ ((\frac{27\times4)^{(\frac{1}{4})}\times 2^{(\frac{2}{6})}}{(3^4)^{(\frac{1}{4})}}))\\ =(\frac{3\times 2^{(\frac{2}{3}) }\times 2^{(\frac{1}{3})}}{3})\\= 2^{{(\frac{2}{3})}+{(\frac{1}{3})}}\\=2$

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

$\dfrac{(625)^{6.25} \times (25)^{2.6}}{(625)^{6.75} \times (5)^{1.2}} = ?$

  1. $5$
  2. $10$
  3. $15$
  4. $25$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given,

$\dfrac{625^{6.25}\cdot \:25^{2.6}}{625^{6.75}\cdot \:5^{1.2}}$

$=\dfrac{625^{\tfrac{25}{4}}\cdot \:25^{\tfrac{13}{5}}}{625^{\tfrac{27}{4}}\cdot \:5^{\tfrac{6}{5}}}$

$=\dfrac{(5^4)^{\tfrac{25}{4}}\cdot \:(5^2)^{\tfrac{13}{5}}}{(5^4)^{\tfrac{27}{4}}\cdot \:5^{\tfrac{6}{5}}}$

$=\dfrac{(5^{25})\cdot \:(5)^{\tfrac{26}{5}}}{(5)^{27}\cdot \:5^{\tfrac{6}{5}}}$

$=\dfrac{5^{25+\tfrac{26}{5}}}{5^{27+\tfrac{6}{5}}}$

$=5^{25+\tfrac{26}{5}-27-\tfrac{6}{5}}$

$=5^{-2+\tfrac{20}{5}}$

$=5^{-2+4}$

$=5^2=25$
Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

$\displaystyle (64)^{-\tfrac{1}{2}}-(-32)^{-\tfrac{4}{5}}=?$

  1. $\displaystyle \frac{1}{8}$
  2. $\displaystyle \frac{3}{8}$
  3. $\displaystyle \frac{1}{16}$
  4. $\displaystyle \frac{3}{16}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle (64) ^{-\tfrac{1}{2}}-(-32)^{-\tfrac{4}{5}}$
$=\left({2^6}\right)^{\frac{-1}{2}}-\left({(-2)^5}\right)^{\frac{-4}{5}}$
$=(2)^{ -\tfrac { 6 }{ 2 }  }-(-2)^{ -\tfrac { 4\times 5 }{ 5 }  }$
$=(2)^{ -3 }-(-2)^{ -4 }$
$\dfrac { 1 }{ 8 } -\dfrac { 1 }{ 16 } =\dfrac { 1 }{ 16 } $
Answer $C$ option, $ \cfrac { 1 }{ 16 } $

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

Find m so that $\displaystyle \left ( \frac{11^{2}}{13^{2}} \right )^{-6}=\left ( \frac{13}{11} \right )^{m}$

  1. $-12$
  2. $-6$
  3. $6$
  4. $12$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \left ( \frac{11^{2}}{13^{2}} \right )^{-6}=\left ( \frac{11}{13} \right )^{-12}=\left ( \frac{13}{11} \right )^{12}$
Also,$\displaystyle \left ( \frac{13}{11} \right )^{12}=\left ( \frac{13}{11} \right )^{m}$
$\displaystyle \therefore m=12$