Lagrange multiplier method.
f(x) = x2y7 fx = 2xy7 fy = 7x2y6
g(x,y) = x2+y2 - 1 gx = 2x gy = 2y
fx - λgx = 0 fy - λgy = 0
2xy7 - λ(2x) = 0 leads to x=0 or λ=y7 the above leads to y6(7x2-2y2) = 0 and eventually to
y=0 or 7x2 = 2(1-x2) and thus x = √2 / 3 so y=√7 / 3.
In the cases where x or y = 0, then z = 0; in the case where (x,y) have the above radical values, z has its maximum,
z = (2/9)(√7 / 3)7 which you may evaluate decimally if desired.