Question

find the cube root of 72-32√5

Answer 1

Well, let's suppose there is a rational number x such that

72 + 32sqrt(5) = [ x+ sqrt(5) ]^3

Then expanding the right-hand side,

72 + 32sqrt(5)
= x^3 +3 x^2 sqrt(5) + 3 x (sqrt(5))^2 + (sqrt(5))^3
= x^3 +3x^2 sqrt(5) + 15x + 5sqrt(5)
= (x^3 + 15x) + (3x^2 + 5)sqrt(5)

Set 3x^2 +5 = 32 and x^3 + 15x = 72.

Looking at the first equation, you can calculate that x=3. Now you can verify that x=3 also works for the 2nd equation. Therefore

72 +32sqrt(5) = (3 + sqrt(5)) ^ 3

So the cube root of 72 +32sqrt(5) = 3 + sqrt(5).