Find the numerical value of $ P: Q $ where $ \mathrm{P}=\left(\frac{x^{m}}{x^{n}}\right)^{m+n-l} \times\left(\frac{x^{n}}{x^{l}}\right)^{n+l-m} \times\left(\frac{x^{l}}{x^{m}}\right)^{l+m-n} $ and$ \mathrm{Q}=\left(x^{1 /(a-b)}\right)^{1 /(a-c)} \times\left(x^{1 /(b-c)}\right)^{1 /(b-a)} \times\left(x^{1 /(c-a)}\right)^{1 /(c-b)} $
where $ a, b, c $ being all different.A. $ 1: 2 $
B. $ 2: 1 $
C. $ 1: 1 $
D. None of these

Given:

\( \mathrm{P}=\left(\frac{x^{m}}{x^{n}}\right)^{m+n-l} \times\left(\frac{x^{n}}{x^{l}}\right)^{n+l-m} \times\left(\frac{x^{l}}{x^{m}}\right)^{l+m-n} \) and

\( \mathrm{Q}=\left(x^{1 /(a-b)}\right)^{1 /(a-c)} \times\left(x^{1 /(b-c)}\right)^{1 /(b-a)} \times\left(x^{1 /(c-a)}\right)^{1 /(c-b)} \)

To do:

We have to find the numerical value of \( P: Q \).

Solution:

We know that,

$\frac{a^m}{a^n}=a^{m-n}$

$a^m \times a^n = a^ {m+n}$

Therefore,

 $\mathrm{P}=(\frac{x^{m}}{x^{n}})^{m+n-l} \times(\frac{x^{n}}{x^{l}})^{n+l-m} \times(\frac{x^{l}}{x^{m}})^{l+m-n}$

$=(x^{m-n})^{m+n-l} \times(x^{n-l})^{n+l-m} \times(x^{l-m})^{l+m-n}$

$=(x)^{(m-n)\times(m+n-l)} \times x^{(n-l)\times(n+l-m)} \times x^{(l-m)\times(l+m-n)}$

$=(x)^{(m^2+mn-ml-mn-n^2+nl)+(n^2+nl-mn-nl-l^2+ml)+(l^2+ml-nl-ml-m^2+mn)}$

$=(x)^{0}$

$=1$

$\mathrm{Q}=(x^{1 /(a-b)})^{1 /(a-c)} \times(x^{1 /(b-c)})^{1 /(b-a)} \times(x^{1 /(c-a)})^{1 /(c-b)}$

$=(x)^{1 /(a-b)\times1 /(a-c)} \times(x)^{1 /(b-c)\times1 /(b-a)} \times(x)^{1 /(c-a)\times1 /(c-b)}$

$=(x)^{\frac{1}{(a-b)\times(a-c)}+\frac{1}{(b-c)\times(b-a)}+\frac{1}{(c-a)\times(c-b)}}$

$=(x)^{\frac{(b-c)}{(a-b)\times(a-c)\times(b-c)}+\frac{(c-a)}{(b-c)\times(b-a)\times(c-a)}+\frac{(a-b)}{(c-a)\times(c-b)\times(a-b)}}$

$=(x)^{\frac{b-c+c-a+a-b}{(a-b)\times(a-c)\times(b-c)}}$

$=(x)^{\frac{0}{(a-b)\times(a-c)\times(b-c)}}$

$=x^0$

$=1$

Hence,

$P:Q=1:1$.