Question

If `x^4 + 1/x^4 = 527,` find the value of `x^3 + 1/x^3`

Answer 1

Given: `x^4 + 1/x^4 = 527,`

We know that,

`(x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2`

So, 

`(x^2 + 1/x^2)^2 = 527 + 2`

`(x^2 + 1/x^2)^2 = 529`

`x^2 + 1/x^2 = +-sqrt529`

`∴x^2 + 1/x^2 = +-23`

Now, finding `x + 1/x,`

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

`(x + 1/x)^2 = 23 + 2`

`(x + 1/x)^2 = 25`

`x + 1/x = +-sqrt25`

∴ `x + 1/x = +-5`

Thus,

Here, a = x, b = `1/x`

`(x)^3 + (1/x)^3 = (x + 1/x)^3 - 3(x)(1/x)(x + 1/x)`   ...[Using a³ + b³ = (a + b)³ − 3ab(a + b)]

`x^3 + 1/x^3 = (5)^3 - 3(1)(5)`

`x^3 + 1/x^3 = 125 - 15`

∴ `x^3 + 1/x^3 = +-110`