Given the sum x + 1/x = 4 what is x^2 + 1/x^2, x^3 + 1/x^3 ?

To calculate the first
sum:

x^2 + 1/x^2

we'll have to
raise to square the given sum:

x + 1/x =
4

We'll raise to square both
sides:

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

We'll
apply the formula of the binomial raised to square:

(a+b)^2
= a^2 + 2ab + b^2

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

We'll hold to the left side only the sum of the
squares:

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

We'll simplify and we'll
get:

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

x^2 + (1/x)^2 =
14

To calculate the sum of the cubes, we'll
apply the formula:

a^3 + b^3 = (a+b)(a^2 - ab +
b^2)

x^3 + 1/x^3 = (x + 1/x)(x^2 - x*1/x +
1/x^2)

We'll simplify and we'll substitute x^2 + 1/x^2 by
the result 14:

x^3 + 1/x^3 = 4*(14 -
1)

x^3 + 1/x^3 =
4*13

x^3 + 1/x^3 =
52