Question

Simplify : ${\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x^c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}$

Answer 1

$\Rightarrow {\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x^c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}$

$\because \frac{a^m}{a^n}=a^{m-n}$

$\Rightarrow {\left(x^{a-b}\right)}^{a^2+ab+b^2}\times {\left(x^{b-c}\right)}^{b^2+bc+c^2}\times {\left(x^{c-a}\right)}^{c^2+ca+a^2}$

$\Rightarrow x^{(a-b)(a^2+ab+b^2)}\times x^{(b-c)(b^2+bc+c^2)}\times x^{(c-a)(c^2+ca+a^2)}$ [$\because (a^b{)}^c=a^{b\times c}$]

$\Rightarrow x^{a^3-b^3}\times x^{b^3-c^3}\times x^{c^3-a^3}$ [$\because (x-y)(x^2+xy+y^2)=x^3-y^3$]

$\Rightarrow x^{a^3-b^3+b^3-c^3+c^3-a^3}$ [$\because a^b\times a^c=a^{b+c}$]

$\Rightarrow x^0=1$