The expression 2-24 has value- lying between

The correct option is C 135 and 136 (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] ...

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Standard XII

Mathematics

Binomial Theorem

The expressio...

Question

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

134 and 135

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135 and 136

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136 and 137

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None of these

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Solution

The correct option is B
135 and 136

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

= 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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The value of the expression $(2 + \sqrt{2})^{4}$ lies between

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The expression (2+√2)4 has value, lying between

The correct option is B 135 and 136 (2+√(2))4 = (√2)4(√2+1)4 = 4[4C0 + 4C1(√2) + 4C2(√2)2 + 4C3(√2)3 +4C4 (√2)4] = 4 [1 + 4 √2 + 4.32.2 + 4.3.21.2.3.2 √2 + 4] = 4[1 + 4 √2 + 12 + 8 √2 + 4] = 4[17 + 12 √2] = 4[17 + 17] = 4[34] = 136

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