Answer
$30a^{2}b^{4}$
Work Step by Step
$\sqrt{18a^{3}b}\sqrt{50ab^{7}}=\qquad$... apply $\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}$
$=\sqrt{18\cdot 50\cdot a^{3}\cdot a\cdot b\cdot b^{7}}\qquad$ ... apply $a^{m}a^{n}=a^{m+n}$
$=\sqrt{9\cdot 2\cdot 2\cdot 25\cdot a^{4}b^{8}}\qquad$ ... express $a^{2n}$ as $(a^{n})^{2}$
$=\sqrt{3^{2}\cdot 2^{2}\cdot 5^{2}\cdot(a^{2})^{2}(b^{4})^{2}}\qquad $.... apply $\sqrt{x}\cdot\sqrt{y}=\sqrt{xy} $and $\sqrt{x^{2}}=|x|$
$=3\cdot 2\cdot 5\cdot a^{2}\cdot b^{4}$
$=30a^{2}b^{4}$
