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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,
$P:Q=1:1$.