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MCQ

If x = \[\sqrt[3]{2 + \sqrt{3}}\] , then \[x^3 + \frac{1}{x^3} =\]

#### Options

2

4

8

9

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#### Solution

Given that . `x = 3sqrt(2+sqrt3)` It can be simplified as

` x^3 = 2+sqrt3`

`1/ x^3 = 1 /(2+sqrt3)`

We know that rationalization factor for `2+sqrt3` is `2- sqrt3`. We will multiply numerator and denominator of the given expression `1/(2+sqrt3)`by `2-sqrt3`, to get

`1/x^3 = 1/(2+sqrt3 ) xx (2-sqrt3)/(2-sqrt3)`

`= (2-sqrt3)/((2)^2 - (sqrt3)^2)`

`= (2-sqrt3)/(4-3)`

`=2-sqrt3`

Therefore,

`x^3 + 1/x^3 = 2 +sqrt3 +2 - sqrt3`

`= 2+2`

`=4`

Concept: Laws of Exponents for Real Numbers

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