Simplify the following:
(z^2-36/z^2) (z^2+2 z-48-(6 z)/z^2)
Combine powers. z/z^2 = z^(1-2):
(z^2-36/z^2) (z^2+2 z-48-6 z^1-2)
1-2 = -1:
(z^2-36/z^2) (z^2+2 z-48-6 z^-1)
Put each term in z^2+2 z-48-6/z over the common denominator z: z^2+2 z-48-6/z = z^3/z+(2 z^2)/z-(48 z)/z-6/z:
z^3/z+(2 z^2)/z-(48 z)/z-6/z (z^2-36/z^2)
z^3/z+(2 z^2)/z-(48 z)/z-6/z = (z^3+2 z^2-48 z-6)/z:
(z^3+2 z^2-48 z-6)/z (z^2-36/z^2)
Put each term in z^2-36/z^2 over the common denominator z^2: z^2-36/z^2 = z^4/z^2-36/z^2:
(z^4/z^2-36/z^2 (z^3+2 z^2-48 z-6))/(z)
z^4/z^2-36/z^2 = (z^4-36)/z^2:
((z^4-36)/z^2 (z^3+2 z^2-48 z-6))/(z)
z^4-36 = (z^2)^2-6^2:
((z^2)^2-6^2 (z^3+2 z^2-48 z-6))/(z^2 z)
Factor the difference of two squares. (z^2)^2-6^2 = (z^2-6) (z^2+6):
((z^2-6) (z^2+6) (z^3+2 z^2-48 z-6))/(z^2 z)
Combine powers. ((z^2-6) (z^2+6) (z^3+2 z^2-48 z-6))/(z^2 z) = z^(-2-1) (z^2-6) (z^2+6) (z^3+2 z^2-48 z-6):
z^(-2-1) (z^2-6) (z^2+6) (z^3+2 z^2-48 z-6)
-2-1 = -3:
Answer: |
| z^-3 (z^2-6) (z^2+6) (z^3+2 z^2-48 z-6)
