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L. H. S. amp;(x^a/x^ b)^a b·(x^b/x^ c)^b c·(x^c/x^ a)^c a=1 amp;=(x^a+b)^a b·(x^b+c)^b c·(x^c+a)^c a amp;=x^a^2 b^2· x^b^2 c^2· x^c^2 a^2 ...

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AM,GM,HM Inequality

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Question

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

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Show that :

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

Simplify :

$(x^{a + b})^{a - b} . (x^{b + c})^{b - c} . (x^{c + 1})^{c - a}$

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}} = 1

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

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

{(\frac{x^{a^{2} + b^{2}}}{x^{a b}})}^{a + b} {(\frac{x^{b^{2} + c^{2}}}{x^{b c}})}^{b + c} {(\frac{x^{c^{2} + a^{2}}}{x^{a c}})}^{a + c} = x^{2 (a^{3} + b^{3} + c^{3})}

{(x^{a - b})}^{a + b} {(x^{b - c})}^{b + c} {(x^{c - a})}^{c + a} = 1

{\{{(x^{a - a^{- 1}})}^{\frac{1}{a - 1}}\}}^{\frac{a}{a + 1}} = x

{(\frac{a^{x + 1}}{a^{y + 1}})}^{x + y} {(\frac{a^{y + 2}}{a^{z + 2}})}^{y + z} {(\frac{a^{z + 3}}{a^{x + 3}})}^{z + x} = 1

{(\frac{3^{a}}{3^{b}})}^{a + b} {(\frac{3^{b}}{3^{c}})}^{b + c} {(\frac{3^{c}}{3^{a}})}^{c + a} = 1

\frac{x^{a + b} \cdot x^{b + c} \cdot x^{c + a}}{(x^{a} \cdot x^{b} \cdot x^{c})^{2}}

{(\frac{x^{a}}{x^{- b}})}^{a - b} . {(\frac{x^{b}}{x^{- c}})}^{b - c} . {(\frac{x^{c}}{x^{- a}})}^{c - a}

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