We can use the identity (x+y+z)(x^2+y^2+z^2-xy-yz-zx) = x^3+y^3+z^3-3xyz to find the value of x^3 + y^3 + z^3 - 3xyz.
Given x^2 + y^2 + z^2 = 83 and xyz = 15, we can use these values to find (x+y+z)^2 as follows:
(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy+yz+zx)
= 83 + 2(xy+yz+zx)
Now, we can use the identity (x+y+z)(x^2+y^2+z^2-xy-yz-zx) = x^3+y^3+z^3-3xyz and substitute the values we have:
(x+y+z)(x^2+y^2+z^2-xy-yz-zx) = x^3+y^3+z^3-3xyz
(x+y+z)(83-xy-yz-zx) = x^3+y^3+z^3-3(15)
(x+y+z)(83-xy-yz-zx) = x^3+y^3+z^3-45
Now, we can substitute (x+y+z)^2 = 83 + 2(xy+yz+zx) to get:
(x+y+z)(83-xy-yz-zx) = x^3+y^3+z^3-45
(x+y+z)(83-xy-yz-zx) = (x+y+z)^3 - 3(x+y+z)(xy+yz+zx) - 45
(x+y+z)(83-xy-yz-zx) = (x+y+z)^3 - 3xyz(x+y+z) - 45
(x+y+z)(83-xy-yz-zx) = (x+y+z)^3 - 45(x+y+z)
Now, we can substitute the value of (x+y+z) as (x+y+z) = (xyz)^(1/3) = 15^(1/3), and simplify to get:
15^(1/3)(83-xy-yz-zx) = 15^(4/3) - 45*15^(1/3)
15^(1/3)(83-xy-yz-zx) = 36*15^(1/3)
Now we can solve for (xy+yz+zx) as follows:
(xy+yz+zx) = (x^2+y^2+z^2 - (x^3+y^3+z^3 - 3xyz))/2
= (83 - ((x+y+z)^3 - 45(x+y+z) + 3xyz))/2
= (83 - 15^(4/3) + 45*15^(1/3))/2
Finally, we can substitute the values of (xy+yz+zx) and xyz into the expression for x^3+y^3+z^3-3xyz to get:
x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
= (15^(1/3))(83-xy-yz-zx)
= (15^(1/3))(83 - (83 - 15^(4/3) + 45*15^(1/3))/2)
= 15^(1/3)(15^(4/3) - 45*15^(1/3))/2
= 27
The value of x^3 + y^3 + z^3 - 3xyz is 27.