Numerator $=2^{m+3}\cdot3^{2m-n}\cdot5^{m+n+3}\cdot2^{n+1}\cdot3^{n+1}$
$=2^{m+n+4}\cdot3^{2m+1}\cdot5^{m+n+3}$ (i)
Denominator $=2^{m+1}\cdot3^{m+1}\cdot2^{n+3}\cdot5^{n+3}\cdot3^m\cdot5^m$
$=2^{m+n+4}\cdot3^{2m+1}\cdot5^{m+n+3}$ (ii)
Given expression $=1$

Unit digit in ${6374}^{1993}=$ Unit digit in ${(4)}^{1793}$
=Unit digit in $[{({4}^{2})}^{896}\times 4]$
=Unit digit in $(6\times 4)=4$
Unit digit in ${(625)}^{317}=$ Unit digit in ${(5)}^{317}=5$
Unit digit in ${(341)}^{491}=$ Unit digit in ${(1)}^{491}=1$
Required digit$=$ Unit digit in $(4\times 5\times1)=0$

Thus, the number ends with $5$ for $n\in N$ and $n$ is odd or even.

$\displaystyle\frac{\displaystyle1+\frac{1}{5}}{\displaystyle1-\frac{1}{5}}-\left[\left(\frac{2}{3}\right)^3\right]^{\displaystyle\frac{1}{3}}-\left[\left(\frac{6}{5}\right)^2\right]^{\displaystyle-\frac{1}{2}}=\frac{\displaystyle\frac{6}{5}}{\displaystyle\frac{4}{5}}-\left(\frac{2}{3}\right)^1-\left(\frac{6}{5}\right)^{-1}=\frac{6}{4}-\frac{2}{3}-\frac{5}{6}=\frac{18-8-10}{12}=0$

1 kilometer = 10$^3$ meter

$1 J = (1 kg) ( 1 m)^2 (1 sec)^{-2}$
$1 x = (\alpha kg) (\beta m)^2 (\gamma sec)^{-2}$
$\displaystyle \therefore \frac{1 J}{1x} = \left( \frac{1}{\alpha} \right) \left( \frac{1}{\beta} \right)^2 (\gamma)^2 = \frac{\gamma^2}{\alpha \beta^2} $
$\displaystyle \therefore 1 J = \frac{\gamma^2}{\alpha \beta^2}$ or $\displaystyle 1 cal = 4.18 \displaystyle \frac{\gamma^2}{\alpha \beta^2}$

One micron is equal to $10^{-6} m $. It is used for measuring micro level things. 

$1$ nanometer $=10^{-9}$ meter and $1$ attometer $=10^{-18}$ meter

$1$ micrometer $=10^{-6}$ meter and $1$ decameter $=10$ meter

$1$ picometer $=10^{-12}$ meter and $1$ centimeter $=10^{-2}$ meter

$1$ picometer $=10^{-12}$ meter and $1$ milimeter $=10^{-3}$ meter

$1$ decimeter $=10^{-1}$ meter and $1$ megameter $=10^{6}$ meter

$1$ micrometer $=10^{-6}$ meter and $1$ kilometer $=10^{3}$ meter

$1$ kilometer $=10^3$ meter and $1$ decimeter $=10^{-1}$ meter

$1$ nanometer $=10^{-9}$ meter and $1$ femtometer $=10^{-15}$ meter

Yotta is the largest decimal unit prefix and yocto is the smallest decimal unit prefix in the metric system. 

$1$ milimeter $=10^{-3}$ meter and $1$ terameter $=10^{12}$ meter

$1$ nanometer $=10^{-9}$ meter and $1$ centimeter $=10^{-2}$ meter

An electronvolt is a unit of energy equal to the work done on an electron accelerated through a potential difference of $1$ volt.

Electron volt, unit of energy commonly used in atomic and nuclear physics, equal to the energy gained by an electron (a charged particle carrying unit electronic charge) when the electrical potential at the electron increases by one volt. The electron volt equals $1.602\times 10^{-12}\,erg$  or  $1.602\times 10^{-19}\,joule$.

Unit of Energy $= kg. m^2/ s^2$

Unit of Force $= kg. m/s^2$

Unit of Length $= m$

Thus, unit of energy in terms of unit of force and length is given by

Unit of Energy $= (kg. m/s^2) \times (m)$

If the units of length and force are increased by four times,

the unit of energy will be:

Unit of Energy $= (4 .kg. m/s^2) \times (4m)$

Unit of Energy $= 16 (kg. m^2/s^2)$

Therefore, if the units of length and force are increased by four times, the unit of energy will change by $16$ times.

${(3.92^2+3(2.1^4)}^\dfrac{1}{6}$

$\displaystyle { 2 }^{ n }-{ 2 }^{ n-1 }=4$
$\displaystyle \therefore \quad { 2 }^{ n-1 }\left( 2-1 \right) =4$
$\displaystyle \therefore \quad { 2 }^{ n-1 }={ 2 }^{ 2 }$
$\displaystyle \therefore \quad n-1=2$
$\displaystyle \therefore \quad n=3$
$\displaystyle \therefore \quad { n }^{ n }={ 3 }^{ 3 }=27$

$\displaystyle \frac { { \left( 3.63 \right)  }^{ 2 }-{ \left( 2.37 \right)  }^{ 2 } }{ 3.63+2.37 } $
$\displaystyle =\frac { \left( 3.63+2.37 \right) \left( 3.63-2.37 \right)  }{ 3.63+2.37 } $
$\displaystyle =3.63-2.37$
$\displaystyle =1.26$

$\displaystyle 2^{100}\div 2^1=2^{100-1}=2^{99}$

$\displaystyle { \left( { x }^{ 6 }.{ y }^{ { -5 }/{ 4 } } \right)  }^{ { -4 }/{ 3 } }$
$\displaystyle ={ x }^{ -6\times { 4 }/{ 3 } }{ y }^{ \dfrac { -5 }{ 4 } \times -\dfrac { 4 }{ 3 }  }$
$\displaystyle ={ x }^{ -8 }{ y }^{ { 5 }/{ 3 } }$