(16.) \( \sqrt{\frac{15 x}{x^{3}}} \)

To simplify the expression: \[ \sqrt{\frac{15x}{x^{3}}} \] let's follow the steps below: ### Step 1: Simplify the Fraction Inside the Square Root First, simplify the fraction \(\frac{15x}{x^{3}}\). Recall that \(x^{3} = x \cdot x \cdot x\), so: \[ \frac{15x}{x^{3}} = \frac{15x}{x \cdot x \cdot x} = \frac{15}{x^{2}} \] ### Step 2: Rewrite the Square Root Now, the expression becomes: \[ \sqrt{\frac{15}{x^{2}}} \] This can be separated into the square root of the numerator and the square root of the denominator: \[ \sqrt{15} \div \sqrt{x^{2}} \] ### Step 3: Simplify the Square Root of the Denominator The square root of \(x^{2}\) is \(x\) (assuming \(x > 0\)): \[ \sqrt{x^{2}} = x \] ### Step 4: Combine the Simplified Parts Putting it all together: \[ \frac{\sqrt{15}}{x} \] ### Final Simplified Form \[ \sqrt{\frac{15x}{x^{3}}} = \frac{\sqrt{15}}{x} \] **Note:** This simplification assumes that \(x > 0\). If \(x\) can be negative, the absolute value should be considered: \[ \sqrt{\frac{15x}{x^{3}}} = \frac{\sqrt{15}}{|x|} \]