Question

If \[x + \frac{1}{x} = 3\] then \[x^6 + \frac{1}{x^6}\] = 

 

Options

• 927

• 414

• 364

• 322

Answer 1

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.