rationalize the denominator and simplify: $\\frac{3\\sqrt{b}}{\\sqrt{b + 2}}$

Explanation:

Step1: Multiply numerator and denominator by $\sqrt{b}$

$$ \frac{3\sqrt{b}(\sqrt{b} + 2)}{3\sqrt{b}\times\sqrt{b}} $$

Step2: Simplify the denominator

Since $\sqrt{b}\times\sqrt{b}=b$, the denominator becomes $3b$.

Step3: Expand the numerator

Using the distributive property $a(b + c)=ab+ac$, where $a = 3\sqrt{b}$, $b=\sqrt{b}$, and $c = 2$, we get $3\sqrt{b}\times\sqrt{b}+3\sqrt{b}\times2=3b + 6\sqrt{b}$.

Step4: Factor the numerator

Factor out a $3$ from $3b + 6\sqrt{b}$ to get $3(b + 2\sqrt{b})$. Then, we can rewrite the fraction as $\frac{3(b + 2\sqrt{b})}{3b}$.

Step5: Cancel out the common factor

Cancel out the common factor of $3$ in the numerator and denominator.

Answer:

$\frac{b + 2\sqrt{b}}{b}$