Show that: (1 x^ a-b )^ 1 a-c (1 x^ b-c )^ 1 b-a (1 x^ c-a )^ 1 c… - Vidyadip

Question

Show that: $\Big(\frac{1}{\text{x}^{\text{a}-\text{b}}}\Big)^{\frac{1}{\text{a}-\text{c}}}\Big(\frac{1}{\text{x}^{\text{b}-\text{c}}}\Big)^{\frac{1}{\text{b}-\text{a}}}\Big(\frac{1}{\text{x}^{\text{c}-\text{a}}}\Big)^{\frac{1}{\text{c}-\text{b}}}=1$

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Answer

$\text{LHS}=\Big(\frac{1}{\text{x}^{\text{a}-\text{b}}}\Big)^{\frac{1}{\text{a}-\text{c}}}\Big(\frac{1}{\text{x}^{\text{b}-\text{c}}}\Big)^{\frac{1}{\text{b}-\text{a}}}\Big(\frac{1}{\text{x}^{\text{c}-\text{a}}}\Big)^{\frac{1}{\text{c}-\text{b}}}$
$=\Big(\text{x}^{\frac{1}{\text{a}-\text{b}}\times\frac{1}{\text{a}-\text{c}}}\Big)\Big(\text{x}^{\frac{1}{\text{b}-\text{c}}\times\frac{1}{\text{b}-\text{a}}}\Big)\Big(\text{x}^{\frac{1}{\text{c}-\text{a}}\times\frac{1}{\text{a}-\text{b}}}\Big)$
$=\Big(\text{x}^{\frac{1}{\text{a}-\text{b}}\times\frac{1}{\text{a}-\text{c}}+\frac{1}{\text{b}-\text{c}}\times\frac{1}{\text{b}-\text{c}}+\frac{1}{\text{c}-\text{a}}\times\frac{1}{\text{c}-\text{b}}}\Big)$
$=\bigg(\text{x}^{-\frac{1}{(\text{a}-\text{b})(\text{c}-\text{a})}-\frac{1}{(\text{b}-\text{c})(\text{a}-\text{b})}-\frac{1}{(\text{c}-\text{a})(\text{b}-\text{c})}}\bigg)$
$=\bigg(\text{x}^{\frac{-(\text{b}-\text{c})-(\text{c}-\text{a})-(\text{a}-\text{b})}{(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{a})}}\bigg)$
$=\bigg(\text{x}^{\frac{-\text{b}+\text{c}-\text{c}+\text{a}-\text{a}+\text{b}}{(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{a})}}\bigg)$
$=\big(\text{x}^0\big)$
$=1$
$=\text{RHS}$

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