Question

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

• A. $a^3+b^3+c^3$
• B. 1
• C. ${(a+b+c)}^3$
• D. 0

Answer 1

Answer: (B)

1

${\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}{\left(\frac{x^b}{x^c}\right)}^{b^2+bc+c^2}{\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}$
$\Rightarrow \frac{x^{a^3+a^{2}b+a{b}^2}}{x^{a^{2}b+a{b}^2+b^3}}\times \frac{x^{b^3+b^{2}c+b{c}^2}}{x^{b^{2}c+b{c}^2+c^3}}\times \frac{x^{c^3+c^{2}a+c{a}^2}}{x^{a{c}^2+c{a}^2+a^3}}$
$\Rightarrow \frac{x^{a^3}\times x^{a^{2}b}\times x^{a{b}^2}}{x^{a^{2}b}\times x^{a{b}^2}\times x^{b^3}}\times \frac{x^{b^3}\times x^{b^{2}c}\times x^{b{c}^2}}{x^{b^{2}c}\times x^{b{c}^2}\times x^{c^3}}\times \frac{x^{c^3}\times x^{c^{2}a}\times x^{c{a}^2}}{x^{a{c}^2}\times x^{c{a}^2}\times x^{a^3}}$
$\Rightarrow 1$