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Question
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`
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