Question

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

Answer 1

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

On further simplification, we have:

$x^{(a+b)(a^2-ab+b^2)}\times x^{(b+c)(b^2-bc+c^2)}\times x^{(c+a)(c^2-ca+a^2)}$
$[\because \frac{1}{a^{-x}}=a^x]$

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

$=x^{a^3+b^3+b^3+c^3+a^3}$
$=x^{2(a^3+b^3+c^3)}$

Hence, simplified value is $x^{2(a^3+b^3+c^3)}$