()) Multiply. Assume \( v \) and \( u \) are greater than or equal to zero, and write your answer in simplest form. \( 5 \sqrt{55 v^{5}} \cdot \sqrt{35 u^{5} v} \)

To multiply the expressions, we start by using the property of square roots that states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Thus, we have: \[ 5 \sqrt{55 v^{5}} \cdot \sqrt{35 u^{5} v} = 5 \sqrt{(55 v^{5}) \cdot (35 u^{5} v)}. \] Next, we multiply the coefficients and the terms under the square root: \[ 55 \cdot 35 = 1925, \] and for the variables: \[ v^{5} \cdot v = v^{6} \quad \text{and} \quad u^{5} \text{ remains as it is.} \] So, we rewrite the expression inside the square root: \[ 5 \sqrt{1925 u^{5} v^{6}}. \] Now, we simplify the square root. We can factor 1925 as \( 25 \cdot 77 \) (since \( 25=5^2 \) is a perfect square): \[ \sqrt{1925} = \sqrt{25 \cdot 77} = 5 \sqrt{77}. \] Now substituting this back into our expression yields: \[ 5 \cdot 5 \sqrt{77} \cdot \sqrt{u^{5} v^{6}} = 25 \sqrt{77} \cdot \sqrt{u^{5}} \cdot \sqrt{v^{6}}. \] The square roots of the variables can further be simplified: \[ \sqrt{u^{5}} = u^{2} \sqrt{u} \quad \text{and} \quad \sqrt{v^{6}} = v^{3}. \] Putting it all together, we get: \[ 25 \cdot v^{3} \cdot u^{2} \cdot \sqrt{77} \cdot \sqrt{u} = 25 u^{2} v^{3} \sqrt{77 u}. \] Thus, the final answer is: \[ \boxed{25 u^{2} v^{3} \sqrt{77 u}}. \]