Add. Assume that all variables represent positive real numbers. \( \sqrt{\frac{175}{\mathrm{x}^{8}}}+\sqrt{\frac{7}{9 \mathrm{x}^{8}}} \) \( \sqrt{\frac{175}{\mathrm{x}^{8}}}+\sqrt{\frac{7}{9 x^{8}}}=\square \) (Type an exact answer, using radicals as needed. Simplify your answer.)

To simplify the expression \( \sqrt{\frac{175}{\mathrm{x}^{8}}} + \sqrt{\frac{7}{9 \mathrm{x}^{8}}} \), let's break it down step by step. Starting with the first term: \[ \sqrt{\frac{175}{\mathrm{x}^{8}}} = \frac{\sqrt{175}}{\sqrt{x^{8}}} = \frac{\sqrt{175}}{x^{4}} = \frac{5\sqrt{7}}{x^{4}} \quad \text{(since } 175 = 25 \times 7 \text{ and } \sqrt{25} = 5\text{)} \] Now, for the second term: \[ \sqrt{\frac{7}{9 \mathrm{x}^{8}}} = \frac{\sqrt{7}}{\sqrt{9} \cdot \sqrt{x^{8}}} = \frac{\sqrt{7}}{3x^{4}} \quad \text{(since } \sqrt{9} = 3\text{)} \] Now, we can combine the two terms: \[ \frac{5\sqrt{7}}{x^{4}} + \frac{\sqrt{7}}{3x^{4}} = \left(5 + \frac{1}{3}\right) \frac{\sqrt{7}}{x^{4}} \] Converting \( 5 \) into a fraction with a common denominator: \[ 5 = \frac{15}{3} \] So: \[ \left(5 + \frac{1}{3}\right) = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \] Thus, the combined expression is: \[ \frac{16}{3} \cdot \frac{\sqrt{7}}{x^{4}} = \frac{16\sqrt{7}}{3x^{4}} \] Therefore, the simplified expression is: \[ \sqrt{\frac{175}{\mathrm{x}^{8}}} + \sqrt{\frac{7}{9 \mathrm{x}^{8}}} = \frac{16\sqrt{7}}{3x^{4}} = \square \]