((16m^3)/(n))^1/4((4n^-1)/(m^2))^1/2= square (Simplify your answer. Type exponential notation with pc

To simplify the expression \((\frac {16m^{3}}{n})^{1/4}(\frac {4n^{-1}}{m^{2}})^{1/2}\), we will follow these steps:<br /><br />1. Simplify each part of the expression separately.<br />2. Combine the simplified parts.<br /><br />### Step 1: Simplify \((\frac {16m^{3}}{n})^{1/4}\)<br /><br />\[<br />(\frac {16m^{3}}{n})^{1/4} = \frac{(16m^3)^{1/4}}{n^{1/4}}<br />\]<br /><br />We know that \(16 = 2^4\), so:<br /><br />\[<br />(16m^3)^{1/4} = (2^4 m^3)^{1/4} = 2^{4 \cdot \frac{1}{4}} m^{3 \cdot \frac{1}{4}} = 2 m^{3/4}<br />\]<br /><br />Thus,<br /><br />\[<br />(\frac {16m^{3}}{n})^{1/4} = \frac{2 m^{3/4}}{n^{1/4}}<br />\]<br /><br />### Step 2: Simplify \((\frac {4n^{-1}}{m^{2}})^{1/2}\)<br /><br />\[<br />(\frac {4n^{-1}}{m^{2}})^{1/2} = \frac{(4n^{-1})^{1/2}}{(m^2)^{1/2}}<br />\]<br /><br />We know that \(4 = 2^2\), so:<br /><br />\[<br />(4n^{-1})^{1/2} = (2^2 n^{-1})^{1/2} = 2^{2 \cdot \frac{1}{2}} n^{-1 \cdot \frac{1}{2}} = 2 n^{-1/2}<br />\]<br /><br />And,<br /><br />\[<br />(m^2)^{1/2} = m^{2 \cdot \frac{1}{2}} = m<br />\]<br /><br />Thus,<br /><br />\[<br />(\frac {4n^{-1}}{m^{2}})^{1/2} = \frac{2 n^{-1/2}}{m}<br />\]<br /><br />### Step 3: Combine the simplified parts<br /><br />Now we multiply the two simplified parts together:<br /><br />\[<br />\frac{2 m^{3/4}}{n^{1/4}} \cdot \frac{2 n^{-1/2}}{m} = \frac{2 \cdot 2 \cdot m^{3/4} \cdot n^{-1/2}}{n^{1/4} \cdot m}<br />\]<br /><br />Combine the constants and the exponents:<br /><br />\[<br />= \frac{4 m^{3/4} n^{-1/2}}{n^{1/4} m}<br />\]<br /><br />Simplify the exponents of \(m\) and \(n\):<br /><br />\[<br />= 4 m^{3/4 - 1} n^{-1/2 - 1/4}<br />\]<br /><br />\[<br />= 4 m^{-1/4} n^{-3/4}<br />\]<br /><br />So, the simplified expression is:<br /><br />\[<br />\boxed{4 m^{-1/4} n^{-3/4}}<br />\]