Divide and, if possible, simplify. Assume that all variables represent positive real numbers. \( \frac{\sqrt{84 \mathrm{x}^{7}}}{\sqrt{3 x^{-3}}} \) \( \frac{\sqrt{84 x^{7}}}{\sqrt{3 x^{-3}}}=\square \) \( ( \) Simplify your answer. Type an exact answer, using radicals as needed.)

To simplify the expression \( \frac{\sqrt{84 x^{7}}}{\sqrt{3 x^{-3}}} \), we can combine the square roots using the property \( \sqrt{a} / \sqrt{b} = \sqrt{a/b} \). First, we simplify the numerator and denominator separately: 1. Combine under a single square root: \[ \frac{\sqrt{84 x^{7}}}{\sqrt{3 x^{-3}}} = \sqrt{\frac{84 x^{7}}{3 x^{-3}}} \] 2. Simplify the fraction inside the square root: \[ \frac{84 x^{7}}{3 x^{-3}} = \frac{84}{3} \cdot \frac{x^{7}}{x^{-3}} = 28 \cdot x^{7 + 3} = 28 x^{10} \] 3. Now, we have: \[ \sqrt{28 x^{10}} \] 4. Simplify this expression: - Factor \( 28 \): \( 28 = 4 \cdot 7 = 2^2 \cdot 7 \) - So we can write: \[ \sqrt{28 x^{10}} = \sqrt{4 \cdot 7 \cdot x^{10}} = \sqrt{4} \cdot \sqrt{7} \cdot \sqrt{x^{10}} = 2 \cdot \sqrt{7} \cdot x^{5} \] Thus, the final simplified result is: \[ 2x^{5}\sqrt{7} \] So, \( \frac{\sqrt{84 x^{7}}}{\sqrt{3 x^{-3}}} = 2 x^5 \sqrt{7} \).