Use the quotient rule to simplify. Assume that the variable represents a positive real number. \( \sqrt[4]{\frac{x^{3}}{256}} \) \( \sqrt[4]{\frac{x^{3}}{256}}=\square \) (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)
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To simplify \( \sqrt[4]{\frac{x^{3}}{256}} \), we can break it down using the quotient rule for radicals. This gives us: \[ \sqrt[4]{\frac{x^{3}}{256}} = \frac{\sqrt[4]{x^{3}}}{\sqrt[4]{256}}. \] Now, \( \sqrt[4]{256} \) is the same as \( \sqrt[4]{(4^4)} = 4 \). For \( \sqrt[4]{x^{3}} \), we can rewrite it as \( x^{\frac{3}{4}} \). Putting it all together, we get: \[ \sqrt[4]{\frac{x^{3}}{256}} = \frac{x^{\frac{3}{4}}}{4}. \] So the final simplified expression is: \[ \frac{x^{\frac{3}{4}}}{4}. \]
