The value of the expression (2+√2)4 lies between

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Question

The value of the expression

(2 + \sqrt{2})^{4}

lies between

Options:

A . 134 and 135

B . 135 and 136

C . 136 and 137

D . None of these

Answer: Option 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 [1 + 4

\sqrt{2}

+

\frac{4.3}{2}

.2 +

\frac{4.3.2}{1.2.3}

.2

\sqrt{2}

+ 4]
= 4[1 + 4

\sqrt{2}

+ 12 + 8

\sqrt{2}

+ 4] = 4[17 + 12

\sqrt{2}

]
= 4[17 + 17] = 4[34] = 136

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