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Question
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^3+b^3+c^3))`
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Solution
`(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^3+b^3+c^3))`
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)(c^2+a^2-ac))]`
`=(x^(a^3+b^3))(x^(b^3+c^3))(x^(a^3+c^3))`
`=x^(2(a^3+b^3+c^3))`
= RHS
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