Question:
Find $(a+b)^{4}-(a-b)^{4}$. Hence, evaluate $(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}$.
Solution:
The expression $(a+b)^{4}-(a-b)^{4}$ can be written as
$(a+b)^{4}-(a-b)^{4}=2\left[{ }^{4} C_{1} a^{3} b^{1}+{ }^{4} C_{3} a^{1} b^{3}\right]$
$=2\left[4 a^{3} b+4 a b^{3}\right]$
$=8\left(a^{3} b+a b^{3}\right)$
Putting $a=\sqrt{3}$ and $b=\sqrt{2}$, we get:
$(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}=8\left[(\sqrt{3})^{3} \times \sqrt{2}+\sqrt{3} \times(\sqrt{2})^{3}\right]$
$=8(3 \sqrt{6}+2 \sqrt{6})$
$=40 \sqrt{6}$
$\therefore(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}=40 \sqrt{6}$