Write the following in simplified radical form. \[ \sqrt[5]{243 y^{14}} \] Assume that the variable represents a positive

To simplify \( \sqrt[5]{243 y^{14}} \), we can break it down into two parts: the numerical and the variable part. First, let's simplify \( 243 \). We know that \( 243 = 3^5 \), so \( \sqrt[5]{243} = \sqrt[5]{3^5} = 3 \). Next, for the variable \( y^{14} \), we can apply the properties of exponents. The fifth root of \( y^{14} \) can be simplified as follows: \[ y^{14} = y^{5 \cdot 2 + 4} = (y^5)^2 \cdot y^4 \] Thus, \( \sqrt[5]{y^{14}} = \sqrt[5]{(y^5)^2 \cdot y^4} = (y^2) \cdot \sqrt[5]{y^4} \). Putting it all together, we have: \[ \sqrt[5]{243 y^{14}} = 3y^2 \sqrt[5]{y^4} \]. So, the final answer in simplified radical form is: \[ 3y^2 \sqrt[5]{y^4} \].