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Question
If \[x + \frac{1}{x} = 3\] then \[x^6 + \frac{1}{x^6}\] =
Options
927
414
364
322
Solution
In the given problem, we have to find the value of `x^6 + 1/x^6`
Given `x+ 1/x =3`
We shall use the identity `(a + b)^3 = a^3 + b^3 + 3ab (a+b)`and `a^2 + b^2 + 2ab = (a+b)`
Here put `x+ 1/x = 3`,
`(x+ 1/x)^2 = x^2 + 1/x^2 + 2( x xx 1/x)`
`(3)^2 = x^2 + 1/x^2 + 2 (x xx 1/x)`
`9 = x^2 + 1/x^2 + 2`
`9-2 = x^2 + 1/x^2`
`7 = x^2 + 1/x^2`
Take Cube on both sides we get,
`(x^2 + 1/x^2 )^3 = (x^2)^3 + 1/(x^2)^3 + 3 (x^2 xx 1/x^2)(x^2 + 1/x^2)`
`(7)^3 = x^6 + 1/x^6 + 3(x^2 xx 1/x^2) (7)`
`343 = x^6 + 1/x^6 + 7 xx 3`
`343 - 21 = x^6 + 1/x^6`
`322 = x^6 + 1/x^6`
Hence the value of `x^6 + 1/x^6` is 322.
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