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Question
Prove that:
`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`
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Solution
Consider the left hand side:
`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)`
`=x^(a(a^2+ab+b^2))/x^(b(a^2+ab+b^2))xxx^(b(b^2+bc+c^2))/x^(c(b^2+bc+c^2))xxx^(c(c^2+ca+a^2))/x^(a(c^2+ca+a^2))`
`=x^(a(a^2+ab+b^2)-b(a^2+ab+b^2))xxx^(b(b^2+bc+c^2)-c(b^2+bc+c^2))xxx^(c(c^2+ca+a^2)-a(c^2+ca+a^2))`
`=x^((a-b)(a^2+ab+b^2))xxx^((b-c)(b^2+bc+c^2))xxx^((c-a)(c^2+ca+a^2))`
`=x^((a^3-b^3))xxx^((b^3-c^3))xxx^((c^3-a^3))`
`=x^((a^3-b^3+b^3-c^3+c^3-a^3))`
`=x^0`
= 1
LHS = RHS
Hence proved.
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