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, ${\left(1+\sqrt{2}\right)}^4-{\left(1-\sqrt{2}\right)}^4$
$=2\left[{}^4{C}_1\sqrt{2}{+}^4{C}_3{\left(\sqrt{2}\right)}^3\right]$
$\left[\because {\left(a+b\right)}^4-{\left(a-b\right)}^4=2\left({}^4{C}_1{a}^{3}b{+}^4{C}_{3}a{b}^3\right)\right]$
$=2\left[4\times \sqrt{2}+4\times 2\sqrt{2}\right]$
$=2\left(12\sqrt{2}\right)$
$=24\sqrt{2}$
Given, $n(A)=3,n(B)=4$