What is the value of the expression ( 1 + 2 ) 4 - ( 1 - 2 ) 4 ? ?

Question

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

Smart Achievers · Accepted Answer

(4)

$Consider, {(1 + \sqrt{2})}^{4} - {(1 - \sqrt{2})}^{4}$

$= 2 [{}^{4}C_{1} \sqrt{2} + {}^{4}C_{3} {(\sqrt{2})}^{3}]$ $[∵ {(a + b)}^{4} - {(a - b)}^{4} = 2 ({}^{4}C_{1} a^{3} b + {}^{4}C_{3} a b^{3})]$

$= 2 [4 \sqrt{2} + 4 \times 2 \sqrt{2}] = 2 (12 \sqrt{2}) = 24 \sqrt{2}$

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