((256m^5)/(n))^1/8((625n^-1)/(m^4))^1/4= square (Simplify your answer.Type exponential notation with p

To simplify the given expression, we need to apply the properties of exponents and simplify each part separately.<br /><br />Given expression:<br />\[<br />\left(\frac{256m^5}{n}\right)^{1/8} \left(\frac{625n^{-1}}{m^4}\right)^{1/4}<br />\]<br /><br />First, let's simplify \(\left(\frac{256m^5}{n}\right)^{1/8}\):<br /><br />\[<br />\left(\frac{256m^5}{n}\right)^{1/8} = \frac{(256m^5)^{1/8}}{n^{1/8}}<br />\]<br /><br />We know that \(256 = 2^8\), so:<br /><br />\[<br />(256m^5)^{1/8} = (2^8 m^5)^{1/8} = 2^{8 \cdot \frac{1}{8}} m^{5 \cdot \frac{1}{8}} = 2 m^{5/8}<br />\]<br /><br />Thus,<br /><br />\[<br />\left(\frac{256m^5}{n}\right)^{1/8} = \frac{2 m^{5/8}}{n^{1/8}}<br />\]<br /><br />Next, let's simplify \(\left(\frac{625n^{-1}}{m^4}\right)^{1/4}\):<br /><br />\[<br />\left(\frac{625n^{-1}}{m^4}\right)^{1/4} = \frac{(625n^{-1})^{1/4}}{(m^4)^{1/4}}<br />\]<br /><br />We know that \(625 = 5^4\), so:<br /><br />\[<br />(625n^{-1})^{1/4} = (5^4 n^{-1})^{1/4} = 5^{4 \cdot \frac{1}{4}} n^{-1 \cdot \frac{1}{4}} = 5 n^{-1/4}<br />\]<br /><br />And,<br /><br />\[<br />(m^4)^{1/4} = m^{4 \cdot \frac{1}{4}} = m<br />\]<br /><br />Thus,<br /><br />\[<br />\left(\frac{625n^{-1}}{m^4}\right)^{1/4} = \frac{5 n^{-1/4}}{m}<br />\]<br /><br />Now, we combine the two simplified parts:<br /><br />\[<br />\left(\frac{2 m^{5/8}}{n^{1/8}}\right) \left(\frac{5 n^{-1/4}}{m}\right)<br />\]<br /><br />Multiply the numerators and denominators:<br /><br />\[<br />\frac{2 m^{5/8} \cdot 5 n^{-1/4}}{n^{1/8} \cdot m} = \frac{10 m^{5/8} n^{-1/4}}{n^{1/8} m}<br />\]<br /><br />Simplify the exponents:<br /><br />\[<br />= \frac{10 m^{5/8} n^{-1/4}}{n^{1/8} m} = 10 m^{5/8 - 1} n^{-1/4 - 1/8}<br />\]<br /><br />Since \(m^{5/8 - 1} = m^{-3/8}\) and \(n^{-1/4 - 1/8} = n^{-3/8}\), we get:<br /><br />\[<br />10 m^{-3/8} n^{-3/8}<br />\]<br /><br />Thus, the simplified form of the given expression is:<br /><br />\[<br />\boxed{10 m^{-3/8} n^{-3/8}}<br />\]