Question

Prove that

${\left(\frac{x^{a^2}}{x^{b^2}}\right)}^{\frac{1}{a+b}}\times {\left(\frac{x^{b^2}}{x^{c^2}}\right)}^{\frac{1}{b+c}}\times {\left(\frac{x^{c^2}}{x^{a^2}}\right)}^{\frac{1}{c+a}}=1$

Answer 1

Let$a={\left(\frac{x^{a^2}}{x^{b^2}}\right)}^{\frac{1}{a+b}}\times {\left(\frac{x^{b^2}}{x^{c^2}}\right)}^{\frac{1}{c+b}}\times {\left(\frac{x^{c^2}}{x^{a^2}}\right)}^{\frac{1}{a+c}}$

$={\left(x^{a^2-b^2}\right)}^{\frac{1}{a+b}}\times {\left(x^{b^2-c^2}\right)}^{\frac{1}{c+b}}\times {\left(x^{c^2-a^2}\right)}^{\frac{1}{a+c}}$

$=x^{\frac{(a+b)(a-b)}{a+b}}\times x^{\frac{(c+b)(b-c)}{c+b}}\times x^{\frac{(a+c)(c-a)}{a+c}}$

$=x^{a-b}\times x^{b-c}\times x^{c-a}$

$=x^{a-b+b-c+c-a}$

$=x^0$

$a=1$

$\therefore {\left(\frac{x^{a^2}}{x^{b^2}}\right)}^{\frac{1}{a+b}}\times {\left(\frac{x^{b^2}}{x^{c^2}}\right)}^{\frac{1}{c+b}}\times {\left(\frac{x^{c^2}}{x^{a^2}}\right)}^{\frac{1}{a+c}}=1$