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

Given:

\( (\frac{a}{b c}+\frac{b}{c a}+\frac{c}{a b})^{2} \)

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,

$(\frac{a}{b c}+\frac{b}{c a}+\frac{c}{a b})^{2}=(\frac{a}{b c})^{2}+(\frac{b}{c a})^{2}+(\frac{c}{a b})^{2}+2 \times \frac{a}{b c} \times \frac{b}{c a}+2 \frac{b}{c a} \times \frac{c}{a b}+2 \frac{c}{a b} \times \frac{a}{b c}$

$=\frac{a^{2}}{b^{2} c^{2}}+\frac{b^{2}}{c^{2} a^{2}}+\frac{c^{2}}{a^{2} b^{2}}+\frac{2}{c^{2}}+\frac{2}{a^{2}}+\frac{2}{b^{2}}$

$=\frac{a^{2}}{b^{2} c^{2}}+\frac{b^{2}}{c^{2} a^{2}}+\frac{c^{2}}{a^{2} b^{2}}+\frac{2}{a^{2}}+\frac{2}{b^{2}}+\frac{2}{c^{2}}$

Hence, $(\frac{a}{b c}+\frac{b}{c a}+\frac{c}{a b})^{2}=\frac{a^{2}}{b^{2} c^{2}}+\frac{b^{2}}{c^{2} a^{2}}+\frac{c^{2}}{a^{2} b^{2}}+\frac{2}{a^{2}}+\frac{2}{b^{2}}+\frac{2}{c^{2}}$. 

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Updated on: 10-Oct-2022

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