Answer
$a)$ $xy^{2}$
$b)$ $2x^{1/6}$
Work Step by Step
$a)$ $\sqrt[5]{x^{3}y^{2}}\sqrt[10]{x^{4}y^{16}}$
Rewrite the radical expressions as powers with rational exponent:
$\sqrt[5]{x^{3}y^{2}}\sqrt[10]{x^{4}y^{16}}=(x^{3}y^{2})^{1/5}(x^{4}y^{16})^{1/10}=...$
Evaluate the powers:
$...=(x^{3/5}y^{2/5})(x^{2/5}y^{8/5})=...$
Evaluate the product:
$...=x^{3/5+2/5}y^{2/5+8/5}=x^{5/5}y^{10/5}=xy^{2}$
$b)$ $\dfrac{\sqrt[3]{8x^{2}}}{\sqrt{x}}$
Rewrite the radical expressions as powers with rational exponent:
$\dfrac{\sqrt[3]{8x^{2}}}{\sqrt{x}}=\dfrac{(8x^{2})^{1/3}}{x^{1/2}}=...$
Evaluate the power in the numerator:
$...=\dfrac{(\sqrt[3]8)x^{2/3}}{x^{1/2}}=\dfrac{2x^{2/3}}{x^{1/2}}=...$
Evaluate the division:
$...=2x^{2/3-1/2}=2x^{1/6}$
