Question

Find ${\left(a+b\right)}^4-{\left(a-b\right)}^4$. Hence, evaluate ${\left(\sqrt{3}+\sqrt{2}\right)}^4-{\left(\sqrt{3}-\sqrt{2}\right)}^4$.

Answer 1

The expression $(a+b{)}^4-(a-b{)}^4$ can be written as

$$\begin{gathered} (a+b{)}^4-(a-b{)}^4=2[{}^{4}C_1{a}^3{b}^1+{}^{4}C_3{a}^1{b}^3] \\ =2[4a^{3}b+4a{b}^3] \\ =8(a^{3}b+a{b}^3) \end{gathered}$$

$$\begin{gathered} \mathrm{Putting}a=\sqrt{3}\mathrm{and}b=\sqrt{2},\mathrm{we}\mathrm{get}: \\ (\sqrt{3}+\sqrt{2}{)}^4-(\sqrt{3}-\sqrt{2}{)}^4=8\left[\right(\sqrt{3}{)}^3\times \sqrt{2}+\sqrt{3}\times \left(\sqrt{2}{)}^3\right] \\ =8(3\sqrt{6}+2\sqrt{6}) \\ =40\sqrt{6} \end{gathered}$$

$\therefore (\sqrt{3}+\sqrt{2}{)}^4-(\sqrt{3}-\sqrt{2}{)}^4=40\sqrt{6}$