Expand (a b -b a )^ 4 . - Vidyadip

Question

Expand $\left(\frac{a}{b}-\frac{b}{a}\right)^{4}$.

✓

Answer

$ \begin{aligned} &\left(\frac{a}{b}-\frac{b}{a}\right)^{4} \\ &={ }^{4} \mathrm{C}_{0}\left(\frac{a}{b}\right)^{4}\left(\frac{b}{a}\right)^{0}-{ }^{4} \mathrm{C}_{1}\left(\frac{a}{b}\right)^{3}\left(\frac{b}{a}\right)^{1}+{ }^{4} \mathrm{C}_{2}\left(\frac{a}{b}\right)^{2}\left(\frac{b}{a}\right)^{2}-{ }^{4} \mathrm{C}_{3}\left(\frac{a}{b}\right)^{1}\left(\frac{b}{a}\right)^{3}+{ }^{4} \mathrm{C}_{4}\left(\frac{a}{b}\right)^{0}\left(\frac{b}{a}\right)^{4} \\ &=\frac{a^{4}}{b^{4}}-4\left(\frac{a^{3}}{b^{3}}\right)\left(\frac{b}{a}\right)+6\left(\frac{a^{2}}{b^{2}}\right)\left(\frac{b^{2}}{a^{2}}\right)-4\left(\frac{a}{b}\right)\left(\frac{b^{3}}{a^{3}}\right)+\frac{b^{4}}{a^{4}} \\ &=\frac{a^{4}}{b^{4}}-\frac{4 a^{2}}{b^{2}}+6-\frac{4 b^{2}}{a^{2}}+\frac{b^{4}}{a^{4}} \end{aligned} $

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