Write the Following in Expanded Form
\[ \left(\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab}\right)^2 \]
Solution:
Using identity:
\[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \]
\[ \left(\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab}\right)^2 \]
\[ = \left(\frac{a}{bc}\right)^2 + \left(\frac{b}{ca}\right)^2 + \left(\frac{c}{ab}\right)^2 + 2\left(\frac{a}{bc}\right)\left(\frac{b}{ca}\right) + 2\left(\frac{b}{ca}\right)\left(\frac{c}{ab}\right) + 2\left(\frac{c}{ab}\right)\left(\frac{a}{bc}\right) \]
\[ = \frac{a^2}{b^2c^2} + \frac{b^2}{c^2a^2} + \frac{c^2}{a^2b^2} + \frac{2}{c^2} + \frac{2}{a^2} + \frac{2}{b^2} \]