The value of the expression ( 2 + √ 2 ) 4 lies between Options: A . &nbsp 134 and 135 B . &nbsp 135 and 136 C . &nbsp 136 and 137 D . &nbsp None of these

Answer: Option B : B ( 2 + √ ( 2 ) ) 4 = ( √ 2 ) 4 ( √ 2 + 1 ) 4 = 4[ 4 C 0 + 4 C 1 ( √ 2 ) + 4 C 2 ( √ 2 ) 2 + 4 C 3 ( √ 2 ) 3 + 4 C 4 ( √ 2 ) 4 ] = 4 [1 + 4 √ 2 + 4.3 2 .2 + 4.3.2 1.2.3 .2 √ 2 + 4] = 4[1 + 4 √ 2 + 12 + 8 √ 2 + 4] = 4[17 + 12 √ 2 ] = 4[17 + 17] = 4[34] = 136

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

Discussion Forum : Binomial Theorem

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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Discussion Forum : Binomial Theorem

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