Rationalize the denominator and simplify. sqrt ((81m)/(5pi )),ngt 0,mgeqslant 0 (9sqrt (5mn))/(5n) (9sqrt (m))/(5n) o (9sqrt (5mn))/(25) (9sqrt (5mn))/(25n^2)

## Step 1: Simplify the Radical Expression<br />### Start by simplifying the expression under the square root. The given expression is $\sqrt{\frac{81m}{5n}}$. We can separate this into two parts: $\sqrt{81m}$ and $\sqrt{5n}$. Since $81$ is a perfect square, we have $\sqrt{81m} = 9\sqrt{m}$.<br />## Step 2: Rationalize the Denominator<br />### To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{5n}$ to eliminate the square root in the denominator. This gives us:<br />\[<br />\frac{9\sqrt{m} \cdot \sqrt{5n}}{\sqrt{5n} \cdot \sqrt{5n}} = \frac{9\sqrt{5mn}}{5n}<br />\]<br />## Step 3: Verify the Simplification<br />### Ensure that the expression is fully simplified and matches one of the provided options. The expression $\frac{9\sqrt{5mn}}{5n}$ is already in its simplest form.