The expression (2 + 2 )^4 has value, lying between - Vidyadip

MCQ

The expression ${(2 + \sqrt 2 )^4}$ has value, lying between

A
$134$ and $135$
✓
$135$ and $136$
C
$136$ and $137$
D
None of these

✓

Answer

Correct option: B.

$135$ and $136$

b
(b) ${(2 + \sqrt 2 )^4} = {(\sqrt 2 )^4}{(\sqrt 2 + 1)^4}$

= $4\,{[^4}{C_0}{ + ^4}{C_1}(\sqrt 2 ){ + ^4}{C_2}{(\sqrt 2 )^2}{ + ^4}{C_3}{(\sqrt 2 )^3}{ + ^4}{C_4}{(\sqrt 2 )^4}]$

= $4\,\left[ {1 + 4\sqrt 2 + \frac{{4.3}}{2}.2 + \frac{{4.3.2}}{{1.2.3}}.2\sqrt 2 + 4} \right]$

= $4\,[1 + 4\sqrt 2 + 12 + 8\sqrt 2 + 4]$ = $4\,[17 + 12\sqrt 2 ]$

= $4\,[17 + (\tilde - 17)]$ = $4\,[\,34] = 136$.

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