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))` - Mathematics

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))`

`(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

Concept: Laws of Exponents for Real Numbers

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Chapter 2 Exponents of Real Numbers
Exercise 2.2 | Q 4.4 | Page 25