The Mean Expectation Box

We are interetsed in computing the expected geometric mean of $x,y,z$ , or $E[(xyz)^{1/3}].$ We are given that these three random variables are independent of each other, which leads to $E[(xyz)^{1/3}] = E[x^{1/3}] \cdot E[y^{1/3}] \cdot E[z^{1/3}]$ and the integral:

$E[(xyz)^{1/3}] = \int_{0}^{a} \int_{0}^{b} \int_{0}^{c} (xyz)^{1/3} \cdot \frac{1}{abc} dz dy dx = \frac{1}{abc}(\int_{0}^{a} x^{1/3} dx)(\int_{0}^{b} y^{1/3} dy)(\int_{0}^{c} z^{1/3} dz) = \frac{1}{abc}(\frac{3}{4} a^{4/3} \cdot \frac{3}{4} b^{4/3} \cdot \frac{3}{4} c^{4/3}) = \frac{27}{64abc} (abc)^{4/3} = \boxed{\frac{27}{64} (abc)^{1/3}}.$