Show that: ( x^ a^2+b^2 x^ ab )^ a+b ( x^ b^2+c^2 x^bc )^ b+c ( x^ c^2+a^2 x^ ac )^ a+c =x^ 2(a^2+b^2+c^2)

LHS= ( x^ a^2+b^2 x^ ab )^ a+b ( x^ b^2+c^2 x^bc )^ b+c ( x^ c^2+a^2 x^ ac )^ a+c = (x^ a^2+b^2-ab )^ a+b (x^ b^2+c^2-bc )^ b+c (x^ c^2+a^2-ac )^ a+c = (x^ (a+b)(a^2+b^2-ab) ) (x^ (b+c)(b^2+c^2-bc) ) (x^ (a+c)(a^2+c^2-ac) ) = (x^ a^3-b^1 ) (x^ b^1-c^1 ) (x^ c^1-a^1 ) =x^ a^1-b^1+b^1-c^1+c^1-a^1 =x^0 =1 =RHS

Show that: ( x^ a^2+b^2 x^ ab )^ a+b ( x^ b^2+c^2 x^bc )^ b+c ( x…

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Question

Show that:$\Big(\frac{\text{x}^{\text{a}^2+\text{b}^2}}{\text{x}^{\text{ab}}}\Big)^{\text{a}+\text{b}}\Big(\frac{\text{x}^{\text{b}^2+\text{c}^2}}{\text{x}^\text{bc}}\Big)^{\text{b}+\text{c}}\Big(\frac{\text{x}^{\text{c}^2+\text{a}^2}}{\text{x}^{\text{ac}}}\Big)^{\text{a}+\text{c}}=\text{x}^{2(\text{a}^2+\text{b}^2+\text{c}^2)}$

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Answer

$\text{LHS}=\Big(\frac{\text{x}^{\text{a}^2+\text{b}^2}}{\text{x}^{\text{ab}}}\Big)^{\text{a}+\text{b}}\Big(\frac{\text{x}^{\text{b}^2+\text{c}^2}}{\text{x}^\text{bc}}\Big)^{\text{b}+\text{c}}\Big(\frac{\text{x}^{\text{c}^2+\text{a}^2}}{\text{x}^{\text{ac}}}\Big)^{\text{a}+\text{c}}$$=\big(\text{x}^{\text{a}^2+\text{b}^2-\text{ab}}\big)^{\text{a}+\text{b}}\big(\text{x}^{\text{b}^2+\text{c}^2-\text{bc}}\big)^{\text{b}+\text{c}}\big(\text{x}^{\text{c}^2+\text{a}^\text{2}-\text{ac}}\big)^{\text{a}+\text{c}}$
$=\big(\text{x}^{(\text{a}+\text{b})(\text{a}^2+\text{b}^2-\text{ab})}\big)\big(\text{x}^{(\text{b}+\text{c})(\text{b}^2+\text{c}^2-\text{bc})}\big)\big(\text{x}^{(\text{a}+\text{c})(\text{a}^2+\text{c}^2-\text{ac})}\big)$
$=\big(\text{x}^{\text{a}^3-\text{b}^1}\big)\big(\text{x}^{\text{b}^1-\text{c}^1}\big)\big(\text{x}^{\text{c}^1-\text{a}^1}\big)$
$=\text{x}^{\text{a}^1-\text{b}^1+\text{b}^1-\text{c}^1+\text{c}^1-\text{a}^1}$
$=\text{x}^0$
$=1$
$=\text{RHS}$