Question

If `x + 1/x = 4`, find the value of `x^3 + 1/x^3`.

Answer 1

Given: `x + 1/x = 4`

Step-wise calculation:

1. Cube both sides using the identity (a + b)³ = a³ + b³ + 3ab(a + b):

`(x + 1/x)^3 = x^3 + 1/x^3 + 3 xx x xx 1/x xx (x + 1/x)`

2. Substitute the given value:

`4^3 = x^3 + 1/x^3 + 3 xx 1 xx 4`

`64 = x^3 + 1/x^3 + 12`

3. Subtract 12 from both sides:

`x^3 + 1/x^3 = 64 - 12`

`x^3 + 1/x^3 = 52`

Thus, the value of `x^3 + 1/x^3` when `x + 1/x = 4` is 52.