Question

Prove that:

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

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

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

Answer 1

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

Consider the left hand side:
$$\begin{gathered} {\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x^c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2} \\ =\frac{x^{a\left(a^2+ab+b^2\right)}}{x^{b\left(a^2+ab+b^2\right)}}\times \frac{x^{b\left(b^2+bc+c^2\right)}}{x^{c\left(b^2+bc+c^2\right)}}\times \frac{x^{c\left(c^2+ca+a^2\right)}}{x^{a\left(c^2+ca+a^2\right)}} \\ =x^{a\left(a^2+ab+b^2\right)-b\left(a^2+ab+b^2\right)}\times x^{b\left(b^2+bc+c^2\right)-c\left(b^2+bc+c^2\right)}\times x^{c\left(c^2+ca+a^2\right)-a\left(c^2+ca+a^2\right)} \\ =x^{\left(a-b\right)\left(a^2+ab+b^2\right)}\times x^{\left(b-c\right)\left(b^2+bc+c^2\right)}\times x^{\left(c-a\right)\left(c^2+ca+a^2\right)} \\ =x^{\left(a^3-b^3\right)}\times x^{\left(b^3-c^3\right)}\times x^{\left(c^3-a^3\right)} \\ =x^{\left(a^3-b^3+b^3-c^3+c^3-a^3\right)} \\ =x^0 \\ =1 \end{gathered}$$
Left hand side is equal to right hand side.
Hence proved.

(ii) ${\left(\frac{x^a}{x^{-b}}\right)}^{a^2-ab+b^2}\times {\left(\frac{x^b}{x^{-c}}\right)}^{b^2-bc+c^2}\times {\left(\frac{x^c}{x^{-a}}\right)}^{c^2-ca+a^2}=1$
Consider the left hand side:
$$\begin{gathered} {\left(\frac{x^a}{x^{-b}}\right)}^{a^2-ab+b^2}\times {\left(\frac{x^b}{x^{-c}}\right)}^{b^2-bc+c^2}\times {\left(\frac{x^c}{x^{-a}}\right)}^{c^2-ca+a^2} \\ =x^{\left(a+b\right)\left(a^2-ab+b^2\right)}\times x^{\left(b+c\right)\left(b^2-bc+c^2\right)}\times x^{\left(c+a\right)\left(c^2-ca+a^2\right)} \\ =x^{\left(a^3+b^3\right)}\times x^{\left(b^3+c^3\right)}\times x^{\left(c^3+a^3\right)} \\ =x^{2\left(a^3+b^3+c^3\right)} \end{gathered}$$
Left hand side is not equal to one.
Disclaimer: The question given in the book is not correct, the left hand side can not be equal to the right hand side.

(iii) ${\left(\frac{x^a}{x^b}\right)}^c\times {\left(\frac{x^b}{x^c}\right)}^a\times {\left(\frac{x^c}{x^a}\right)}^b=1$
Consider the left hand side:
$$\begin{gathered} =\frac{x^{ac}}{x^{bc}}\times \frac{x^{ba}}{x^{ca}}\times \frac{x^{cb}}{x^{ab}} \\ =\frac{x^{ac}}{x^{bc}}\times \frac{x^{ba}}{x^{ca}}\times \frac{x^{cb}}{x^{ab}} \\ =\frac{x^{ac}\times x^{ba}\times x^{cb}}{x^{bc}\times x^{ca}\times x^{ab}} \\ =\frac{x^{ac+ba+cb}}{x^{bc+ca+ab}} \\ =1 \end{gathered}$$
Left hand side is equal to right hand side.
Hence proved.