Question

State true or false

Simplified form of $(\frac{x^a}{x^b}{)}^{a^2+ab+b^2}\times (\frac{x^b}{x^c}{)}^{b^2+bc+c^2}\times (\frac{x^c}{x^a}{)}^{c^2+ca+a^2}$ is 1

• A. True
• B. False

Answer 1

The correct option is A True

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

$\Rightarrow \frac{x^{a.a^2+ab+b^2}}{x^{b.a^2+ab+b^2}}\times \frac{x^{b.b^2+bc+c^2}}{x^{c.b^2+bc+c^2}}\times \frac{x^{c.c^2+ca+a^2}}{x^{a.c^2+ca+a^2}}$

$\Rightarrow \frac{x^{a^3+a^{2}b+a{b}^2}}{x^{a^{2}b+a{b}^2+b^3}}\times \frac{x^{b^3+b^{2}c+b{c}^2}}{x^{c{b}^2+b{c}^2+c^3}}\times \frac{x^{c^3+c^{2}a+c{a}^2}}{x^{a{c}^2+a^{2}c+a^3}}$

$\Rightarrow \frac{x^{a^3+a^{2}b+a{b}^2+b^3+b^{2}c+b{c}^2+c^3+c^{2}a+c{a}^2}}{x^{b{a}^2+a{b}^2+b^3+c{b}^2+b{c}^2+c^3+a{c}^2+c{a}^2+a^3}}$

$\Rightarrow x^{a^3+a^{2}b+a{b}^2+b^3+b^{2}c+b{c}^2+c^3+c^{2}a+c{a}^2-(b{a}^2+a{b}^2+b^3+c{b}^2+b{c}^2+c^3+a{c}^2+c{a}^2+a^3)}$

$\Rightarrow x^0\Rightarrow 1$