Write the following expression in simplified radical form. \[ \sqrt[4]{80 x^{8} y^{15}} \] Assume that all of the variables in the expression represent positive real numbers. \[ \square \] \[ \sqrt{\square} \sqrt[\square]{\square} \]

Step 1 ：The given expression is a fourth root. To simplify it, we need to find the factors of the number and the variables that are perfect fourth powers.

Step 2 ：For the number 80, the perfect fourth power factor is 16 (since \(2^4 = 16\)).

Step 3 ：For the variable x, \(x^8\) is a perfect fourth power since \((x^2)^4 = x^8\).

Step 4 ：For the variable y, \(y^{12}\) is a perfect fourth power since \((y^3)^4 = y^{12}\).

Step 5 ：The remaining factors, 5 and \(y^3\), cannot be simplified further.

Step 6 ：Therefore, the simplified radical form of the expression is \(2x^2y^3\) times the fourth root of \(5y^3\).

Step 7 ：\(\boxed{2x^{2}y^{3}\sqrt[4]{5y^{3}}}\) is the final answer.