Question

simplify.
\sqrt{3b^3} \sqrt{15b}
assume that the variable represents a positive real number.

Explanation:

Step1: Multiply the radicands

Using the property $\sqrt{a}\sqrt{b}=\sqrt{ab}$, we have $\sqrt{3b^{3}}\sqrt{15b}=\sqrt{(3b^{3})(15b)}$.
Calculating the product inside the square root: $(3b^{3})(15b)=3\times15\times b^{3}\times b = 45b^{4}$.
So now we have $\sqrt{45b^{4}}$.

Step2: Simplify the square root

Factor $45$ as $9\times5$ and $b^{4}$ as $(b^{2})^{2}$. Then $\sqrt{45b^{4}}=\sqrt{9\times5\times(b^{2})^{2}}$.
Using the property $\sqrt{ab}=\sqrt{a}\sqrt{b}$ again, we get $\sqrt{9}\times\sqrt{5}\times\sqrt{(b^{2})^{2}}$.
Since $\sqrt{9} = 3$ and $\sqrt{(b^{2})^{2}}=b^{2}$ (because $b$ is a positive real number), we have $3\times b^{2}\times\sqrt{5}$.

Step3: Combine the terms

Combining the terms, we get $3b^{2}\sqrt{5}$.

Answer:

$3b^{2}\sqrt{5}$