Simplify. Assume \( x \) and \( y \) are greater than or equal to zero \( 9 \sqrt{294 x^{7} y^{9}} \)

To simplify \( 9 \sqrt{294 x^{7} y^{9}} \), we start by breaking down the square root: 1. **Factor out the square root**: We can write \( \sqrt{294} \) as \( \sqrt{49 \cdot 6} = 7 \sqrt{6} \) since \( 49 \) is a perfect square. 2. **Extract powers of \(x\) and \(y\)**: - For \(x^{7}\), we can extract \(x^{6}\) as \( (x^{3})^2 \) and leave \(x\) inside the root, so: \(\sqrt{x^7} = \sqrt{x^6 \cdot x} = x^3 \sqrt{x}\). - For \(y^{9}\), we extract \(y^{8}\) as \( (y^{4})^2 \) and leave \(y\) inside the root, so: \(\sqrt{y^9} = \sqrt{y^8 \cdot y} = y^4 \sqrt{y} \). 3. **Putting it all together**: Therefore, we can combine all of this: \[ 9 \sqrt{294 x^{7} y^{9}} = 9 \cdot 7 \sqrt{6} \cdot x^3 \sqrt{x} \cdot y^4 \sqrt{y} = 63 x^3 y^4 \sqrt{6xy}. \] The final simplified form is: \[ 63 x^3 y^4 \sqrt{6xy}. \]