What is x^2+x^-2, if x-(1/x)=4?

First, we'll apply the rule of negative power for the term
x^-2:

x^-2 = 1/x^2

To get the
squares x^2 and (1/x^2), we'll raise to square the expression x-
(1/x).

If we want to raise to square x- (1/x), we'll apply
the formula:

(x-y)^2 = x^2 - 2xy +
y^2

But  x- (1/x) = 4

[x-
(1/x)]^2 = 4^2

x^2 -2*x*(1/x) + 1/x^2 =
16

We'll simplify the ratio 2*x*(1/x) =
2

x^2 - 2 + 1/x^2 = 16

We'll
add 2 both sides:

x^2 + 1/x^2 = 16 +
2

The requested sum is: x^2 + 1/x^2 =
18.