Add. Assume that all variables represent positive real numbers. \[ \begin{array}{l}5 \sqrt{3 x^{3}}+2 x \sqrt{27 x} \\ 5 \sqrt{3 x^{3}}+2 x \sqrt{27 x}= \\ \text { (Type an exact answer, using radicals as needed. Simplify your answer.) }\end{array} \]

To simplify the expression \( 5 \sqrt{3 x^{3}} + 2 x \sqrt{27 x} \), let's break it down step-by-step: 1. **First Term:** The first term is \( 5 \sqrt{3 x^{3}} \). We can simplify \( x^{3} \) under the square root: \[ \sqrt{3 x^{3}} = \sqrt{3} \sqrt{x^{3}} = \sqrt{3} x^{\frac{3}{2}} = \sqrt{3} x^{1.5} \quad \text{or} \quad \sqrt{3} x \sqrt{x}. \] Therefore, \[ 5 \sqrt{3 x^{3}} = 5 \sqrt{3} x^{\frac{3}{2}}. \] 2. **Second Term:** The second term is \( 2 x \sqrt{27 x} \). Simplifying \( \sqrt{27 x} \): \[ \sqrt{27 x} = \sqrt{27} \sqrt{x} = 3 \sqrt{3} \sqrt{x}. \] So, \[ 2 x \sqrt{27 x} = 2 x \cdot 3 \sqrt{3} \sqrt{x} = 6 \sqrt{3} x \sqrt{x} = 6 \sqrt{3} x^{1.5} = 6 \sqrt{3} x^{\frac{3}{2}}. \] 3. **Combine the Terms:** Now we combine the two terms: \[ 5 \sqrt{3} x^{\frac{3}{2}} + 6 \sqrt{3} x^{\frac{3}{2}} = (5 + 6) \sqrt{3} x^{\frac{3}{2}} = 11 \sqrt{3} x^{\frac{3}{2}}. \] Putting it together, the simplified expression is: \[ \boxed{11 \sqrt{3} x^{\frac{3}{2}}} \]