Show that: [ x^ a(a-b) x^ a(a+b) x^ b(b-a) x^ b(b+a) ]^ a+b =-3 2 - Vidyadip

Question

Show that:

$\Big[\Big\{\frac{\text{x}^{\text{a}(\text{a}-\text{b})}}{\text{x}^{\text{a}(\text{a}+\text{b})}}\Big\}\div\Big\{\frac{\text{x}^{\text{b}(\text{b}-\text{a})}}{\text{x}^{\text{b}(\text{b}+\text{a})}}\Big\}\Big]^{\text{a}+\text{b}}=-\frac{3}{2}$

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Answer

$\Big[\Big\{\frac{\text{x}^{\text{a}(\text{a}-\text{b})}}{\text{x}^{\text{a}(\text{a}+\text{b})}}\Big\}\div\Big\{\frac{\text{x}^{\text{b}(\text{b}-\text{a})}}{\text{x}^{\text{b}(\text{b}+\text{a})}}\Big\}\Big]^{\text{a}+\text{b}}=1$

$\text{LHS}=\Big[\Big\{\frac{\text{x}^{\text{a}(\text{a}-\text{b})}}{\text{x}^{\text{a}(\text{a}+\text{b})}}\Big\}\div\Big\{\frac{\text{x}^{\text{b}(\text{b}-\text{a})}}{\text{x}^{\text{b}(\text{b}+\text{a})}}\Big\}\Big]^{\text{a}+\text{b}}$

$\Big[\frac{\text{x}^{\text{a}(\text{a}-\text{b})}}{\text{x}^{\text{a}(\text{a}+\text{b})}}\times\frac{\text{x}^{\text{b}(\text{b}-\text{a})}}{\text{x}^{\text{b}(\text{b}+\text{a})}}\Big]^{\text{a}+\text{b}}$

$=\Big[\frac{\text{x}^{\text{a}^2-\text{ab}}}{\text{x}^{\text{a}^2-\text{ab}}}\times\frac{\text{x}^{\text{b}^2+\text{ab}}}{\text{x}^{\text{b}^2-\text{ab}}}\Big]^{\text{a}+\text{b}}$

$=\Big[\frac{\text{x}^{\text{a}^2-\text{ab}+\text{b}^2+\text{ab}}}{\text{x}^{\text{a}^2+\text{ab}+\text{b}^2-\text{ab}}}\Big]^{\text{a}+\text{b}}$

$=\Big[\frac{\text{x}^{\text{a}^2+\text{b}^2}}{\text{x}^{\text{a}^2+\text{b}^2}}\Big]^{\text{a}+\text{b}}$

$=[1]^{\text{a}+\text{b}}$

$=1$

$=\text{RHS}.$

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