Write the following in the expanded form:$ (\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2} $

Given:

\( (\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2} \)

To do:

We have to write the given expression in expanded form.

Solution:

We know that,

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$

Therefore,

$(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2}=(\frac{x}{y})^{2}+(\frac{y}{z})^{2}+(\frac{z}{x})^{2}+2 \times \frac{x}{y} \times \frac{y}{z}+2 \times \frac{y}{z} \times \frac{z}{x}+2 \times \frac{z}{x} \times \frac{x}{y}$

$=\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}+\frac{z^{2}}{x^{2}}+2 \frac{x}{z}+2 \frac{y}{x}+2 \frac{z}{y}$

Hence, $(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^{2}=\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}+\frac{z^{2}}{x^{2}}+2 \frac{x}{z}+2 \frac{y}{x}+2 \frac{z}{y}$.