21. \( 5 \sqrt{2 x} \cdot 3 \sqrt{7 x^{2} y^{3}} \)

To simplify \( 5 \sqrt{2 x} \cdot 3 \sqrt{7 x^{2} y^{3}} \), we start by multiplying the coefficients and then the square roots. First, we can multiply the coefficients: \[ 5 \cdot 3 = 15 \] Next, we simplify the square roots: \[ \sqrt{2x} \cdot \sqrt{7x^{2}y^{3}} = \sqrt{(2x)(7x^{2}y^{3})} = \sqrt{14x^{3}y^{3}} \] Now we combine everything: \[ 15 \sqrt{14x^{3}y^{3}} \] We can further simplify the square root: \[ \sqrt{14x^{3}y^{3}} = \sqrt{14} \cdot \sqrt{x^{3}} \cdot \sqrt{y^{3}} = \sqrt{14} \cdot x^{1.5} \cdot y^{1.5} = \sqrt{14} \cdot x \sqrt{x} \cdot y \sqrt{y} \] Thus, the final expression is: \[ 15 \sqrt{14} x \sqrt{x} y \sqrt{y} \] Overall, the simplified expression is: \[ 15 \sqrt{14} xy \sqrt{xy} \]