Given xgt 0 , rewrite the following expression in simplest radical form. sqrt [4](16x)cdot sqrt [3](8x^2) Answer Attemptiout of 2 square

To simplify the expression $\sqrt [4]{16x}\cdot \sqrt [3]{8x^{2}}$, we need to rewrite each radical in terms of fractional exponents and then combine them.<br /><br />First, let's rewrite each radical:<br />\[<br />\sqrt[4]{16x} = (16x)^{1/4}<br />\]<br />\[<br />\sqrt[3]{8x^2} = (8x^2)^{1/3}<br />\]<br /><br />Next, we express 16 and 8 as powers of 2:<br />\[<br />16 = 2^4 \quad \text{and} \quad 8 = 2^3<br />\]<br /><br />Substituting these into the expression, we get:<br />\[<br />(16x)^{1/4} = (2^4 x)^{1/4} = 2^{4 \cdot \frac{1}{4}} x^{1/4} = 2 x^{1/4}<br />\]<br />\[<br />(8x^2)^{1/3} = (2^3 x^2)^{1/3} = 2^{3 \cdot \frac{1}{3}} x^{2 \cdot \frac{1}{3}} = 2 x^{2/3}<br />\]<br /><br />Now, we multiply the two simplified expressions:<br />\[<br />2 x^{1/4} \cdot 2 x^{2/3}<br />\]<br /><br />Combine the coefficients and the exponents of \(x\):<br />\[<br />2 \cdot 2 \cdot x^{1/4 + 2/3}<br />\]<br /><br />Calculate the product of the coefficients:<br />\[<br />4 \cdot x^{1/4 + 2/3}<br />\]<br /><br />To add the exponents, find a common denominator:<br />\[<br />\frac{1}{4} = \frac{3}{12} \quad \text{and} \quad \frac{2}{3} = \frac{8}{12}<br />\]<br />\[<br />\frac{3}{12} + \frac{8}{12} = \frac{11}{12}<br />\]<br /><br />So, the expression becomes:<br />\[<br />4 x^{11/12}<br />\]<br /><br />Thus, the simplified form of the given expression is:<br />\[<br />\boxed{4 x^{11/12}}<br />\]