Question

Prove that :

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

(ii) ${\left(\frac{x^a}{x^b}\right)}^c\times {\left(\frac{x^b}{x^c}\right)}^a\times {\left(\frac{x^c}{x^a}\right)}^b=1$

Answer 1

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

$LHS={\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x^c}\right)}^{b^2+ab+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}=(x^{a-b}{)}^{a^2+ab+b^2}\times (x^{b-c}{)}^{b^2+bc+c^2}\times (x^{c-a}{)}^{c^2+ca+a^2}=x^{(a-b)\left(a^2+ab+b^2\right)}\times x^{(b-c)\left(b^2+bc+c^2\right)}\times x^{(c-a)(c^2+ca+a^2)}=x^{a^3-b^3}.x^{b^3-c^3}.x^{c^3-a^3}=x^{a^3-b^3+b^3-c^3+c^3-a^3}=x^0=1=RHS$

(ii) ${\left(\frac{x^a}{x^b}\right)}^c\times {\left(\frac{x^b}{x^c}\right)}^a\times {\left(\frac{x^c}{x^a}\right)}^b=1$

$LHS={\left(\frac{x^a}{x^b}\right)}^c\times {\left(\frac{x^b}{x^c}\right)}^a\times {\left(\frac{x^c}{x^a}\right)}^b=(x^{(a-b)}{)}^c\times (x^{(b-c)}{)}^a\times (x^{(c-a)}{)}^b=x^{(a-b)c}\times x^{(b-c)a}\times x^{(c-a)b}=x^{ac-bc}\times x^{ab-ac}\times x^{bc-ab}=x^{ac-bc+ab-ac+bc-ab}=1=RHS$