Combine Radicals Fractional Exponents (L2) Question If the expression x^(1)/(2)y^-(3)/(2)sqrt [3](64x^3y^5) is written in the form ax^by^c then what is the product of a, 5 and c?

To simplify the expression $x^{\frac {1}{2}}y^{-\frac {3}{2}}\sqrt [3]{64x^{3}y^{5}}$, we need to rewrite it in the form $ax^{b}y^{c}$.<br /><br />First, let's rewrite the expression using fractional exponents:<br />$x^{\frac {1}{2}}y^{-\frac {3}{2}}(64x^{3}y^{5})^{\frac {1}{3}}$<br /><br />Now, let's simplify the expression inside the parentheses:<br />$64x^{3}y^{5} = 64x^{3}y^{5}$<br /><br />Next, let's apply the exponent $\frac {1}{3}$ to each term inside the parentheses:<br />$(64x^{3}y^{5})^{\frac {1}{3}} = 64^{\frac {1}{3}}x^{3\cdot\frac {1}{3}}y^{5\cdot\frac {1}{3}}$<br /><br />Simplifying further, we get:<br />$64^{\frac {1}{3}}x^{1}y^{\frac {5}{3}}$<br /><br />Now, let's multiply the terms outside the parentheses with the simplified expression:<br />$x^{\frac {1}{2}}y^{-\frac {3}{2}} \cdot 64^{\frac {1}{3}}x^{1}y^{\frac {5}{3}}$<br /><br />Combining the exponents of $x$ and $y$, we get:<br />$x^{\frac {1}{2}+1}y^{-\frac {3}{2}+\frac {5}{3}}$<br /><br />Simplifying the exponents, we have:<br />$x^{\frac {3}{2}}y^{\frac {1}{6}}$<br /><br />Therefore, the expression can be written in the form $ax^{b}y^{c}$ where $a = 1$, $b = \frac {3}{2}$, and $c = \frac {1}{6}$.<br /><br />The product of $a$, $5$, and $c$ is:<br />$1 \cdot 5 \cdot \frac {1}{6} = \frac {5}{6}$<br /><br />So, the product of $a$, $5$, and $c$ is $\frac {5}{6}$.