Question

What is the value of the expression $ \left(1+\sqrt{2}\right)^{4}-\left(1-\sqrt{2}\right)^{4} $ ?

• A. $ 34 $
• B. $ 10\sqrt{2} $
• C. $ 12+8\sqrt{2} $
• D. $ 24\sqrt{2} $

Answer 1

Answer: (D)

Consider, $\left(1+\sqrt{2}\right)^{4} -\left(1-\sqrt{2}\right)^{4}$
$=2\left[^{4}C_{1}\sqrt{2}+^{4}C_{3}\left(\sqrt{2}\right)^{3}\right]$
$\left[\because \left(a+b\right)^{4}-\left(a-b\right)^{4}=2\left(^{4}C_{1}a^{3}b+^{4}C_{3}ab^{3}\right)\right]$
$=2\left[4\times\sqrt{2}+4\times2\sqrt{2}\right]$
$=2\left(12\sqrt{2}\right)$
$=24\sqrt{2}$
Given, $n(A)=3, n(B)=4$