If y varies directly as the square of x and inversely as the cube root of z, where y = 9 when x = 3 and z = 8, find the equation of variation.

Step 1 ：Step 1: Since y varies directly as the square of x and inversely as the cube root of z, we can express this as \( y = k \frac{x^2}{{z^{1/3}}} \) where k is a constant.

Step 2 ：Step 2: Substitute y = 9, x = 3, z = 8 into the equation to find the constant k. So, \( 9 = k \frac{3^2}{{8^{1/3}}} \)

Step 3 ：Step 3: Solve the equation for k. We find \( k = 9 \times \frac{{8^{1/3}}}{9} = 8^{1/3} \)

Step 4 ：Step 4: Substituting k back into the equation, we get \( y = 8^{1/3} \frac{x^2}{{z^{1/3}}} \)