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(a + b)^2 Expansion and Visualisation

Write the fol...

Question

Write the following in the expanded form:

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

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Solution

Given $(\frac{a}{b c} + \frac{b}{c a} + \frac{c}{a b})^{2}$
Using Identity,
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$∴ (\frac{a}{b c} + \frac{b}{c a} + \frac{c}{a b})^{2}$
$= (\frac{a}{b c})^{2} + (\frac{b}{c a})^{2} + (\frac{c}{a b})^{2} + 2 \times \frac{a}{b c} \times \frac{b}{c a} + 2 \times \frac{b}{c a} \times \frac{c}{a b} + 2 \times \frac{c}{a b} \times \frac{a}{b c}$
$= (\frac{a}{b c})^{2} + (\frac{b}{c a})^{2} + (\frac{c}{a b})^{2} + 2 \times \frac{1}{c^{2}} + 2 \times \frac{1}{a^{2}} + 2 \times \frac{1}{b^{2}}$
$= \frac{a^{2}}{b^{2} c^{2}} + \frac{b^{2}}{c^{2} a^{2}} + \frac{c^{2}}{a^{2} b^{2}} + \frac{2}{c^{2}} + \frac{2}{a^{2}} + \frac{2}{b^{2}}$

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

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(a + b)^2 Expansion and Visualisation

Standard VII Mathematics

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