what is the following product? assume $xgeq0$ and $ygeq0$.\n$sqrt{5x^{8}y^{2}}cdotsqrt{10x^{3}}cdotsqrt{12y}$…

Explanation:

Step1: Combine the square - roots

By the property $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$, we have $\sqrt{5x^{8}y^{2}}\cdot\sqrt{10x^{3}}\cdot\sqrt{12y}=\sqrt{5x^{8}y^{2}\cdot10x^{3}\cdot12y}$.

Step2: Multiply the coefficients and add the exponents of like - bases

First, multiply the coefficients: $5\times10\times12 = 600$. For the $x$ terms, using the rule $a^{m}\cdot a^{n}=a^{m + n}$, we have $x^{8}\cdot x^{3}=x^{8 + 3}=x^{11}$. For the $y$ terms, $y^{2}\cdot y=y^{2+1}=y^{3}$. So the expression becomes $\sqrt{600x^{11}y^{3}}$.

Step3: Simplify the square - root

We can rewrite $600$ as $100\times6$, $x^{11}$ as $x^{10}\cdot x$, and $y^{3}$ as $y^{2}\cdot y$. Then $\sqrt{600x^{11}y^{3}}=\sqrt{100\times6\times x^{10}\cdot x\times y^{2}\cdot y}$. Using the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ ($a = 100x^{10}y^{2}$, $b = 6xy$), we get $\sqrt{100x^{10}y^{2}}\cdot\sqrt{6xy}$. Since $\sqrt{100x^{10}y^{2}} = 10x^{5}y$, the simplified form is $10x^{5}y\sqrt{6xy}$.

Answer:

$10x^{5}y\sqrt{6xy}$ (corresponding to the second option)