Question

For any positive real number x, find the value of ${\left(\frac{x^a}{x^b}\right)}^{a+b}\times {\left(\frac{x^b}{x^c}\right)}^{b+c}\times {\left(\frac{x^c}{x^a}\right)}^{c+a}$

Answer 1

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