Write the following in the expanded form: ( a bc + b ca + c ab )^…

Question

Write the following in the expanded form: $\Big(\frac{\text{a}}{\text{bc}}+\frac{\text{b}}{\text{ca}}+\frac{\text{c}}{\text{ab}}\Big)^2$

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Answer

We have, $\Big(\frac{\text{a}}{\text{bc}}+\frac{\text{b}}{\text{ca}}+\frac{\text{c}}{\text{ab}}\Big)^2=\Big(\frac{\text{a}}{\text{bc}}\Big)^2+\Big(\frac{\text{b}}{\text{ca}}\Big)^2+\Big(\frac{\text{c}}{\text{ab}}\Big)^2$
$+2\Big(\frac{\text{a}}{\text{bc}}\Big)\Big(\frac{\text{b}}{\text{ca}}\Big)+2\Big(\frac{\text{b}}{\text{ca}}\Big)\Big(\frac{\text{c}}{\text{ab}}\Big)+2\Big(\frac{\text{a}}{\text{bc}}\Big)\Big(\frac{\text{c}}{\text{ab}}\Big)$
$\big[\therefore\big(\text{x}+\text{y}+\text{z}\big)=\text{x}^2+\text{y}^2+\text{z}^2+2\text{xy}+2\text{yz}+2\text{xz}\big]$
$\Big(\frac{\text{a}}{\text{bc}}+\frac{\text{b}}{\text{ca}}+\frac{\text{c}}{\text{ab}}\Big)^2\Big(\frac{\text{a}}{\text{bc}}+\frac{\text{b}}{\text{ca}}+\frac{\text{c}}{\text{ab}}\Big)^2$
$=\Big(\frac{\text{a}^2}{\text{b}^2\text{c}^2}\Big)+\Big(\frac{\text{b}^2}{\text{c}^2\text{a}^2}\Big)+\Big(\frac{\text{c}^2}{\text{a}^2\text{b}^2}\Big)+2\frac{2}{\text{a}^2}+\frac{2}{\text{b}^2}+\frac{2}{\text{c}^2}$

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