Solve Fifth Root Expression: Multiplying ∛(3³) and ∛(3²)

In order to simplify the given expression, we will use two laws of exponents:

a. The root law (expanded):

$\sqrt[\textcolor{blue}{n}]{a^{\textcolor{red}{m}}}=a^{\frac{\textcolor{red}{m}}{\textcolor{blue}{n}}} =(\sqrt[\textcolor{blue}{n}]{a})^{\textcolor{red}{m}}$

$$b. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

We'll start by converting the roots to exponent notation using the law of exponents mentioned in a':

$\sqrt[\textcolor{blue}{5}]{3^{\textcolor{red}{3}}}\cdot\sqrt[\textcolor{blue}{5}]{3^{\textcolor{red}{2}}}= \\ \downarrow\\ 3^{\frac{\textcolor{red}{3}}{\textcolor{blue}{5}}}\cdot3^{\frac{\textcolor{red}{2}}{\textcolor{blue}{5}}} =$

We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b':

$3^{\frac{3}{5}}\cdot3^{\frac{2}{5}}= \\ 3^{\frac{3}{5}+\frac{2}{5}}=$

We'll continue and perform (separately) the addition of fractions in the exponent of the expression we got, we'll do this by expanding each of the fractions to the common denominator - the number 5, then we'll perform the multiplication and addition operations in the fraction numerator:

$\frac{3}{5}+\frac{2}{5}=\\ \frac{3+2}{5}=\\ \frac{5}{5}=\\ 1$

Therefore, we get:

$3^{\frac{3}{5}+\frac{2}{5}}=\\ 3^{1}=\\ \boxed{3}$

Let's summarize the expression simplification process:

$\sqrt[5]{3^3}\cdot\sqrt[5]{3^2}= \\ \downarrow\\ 3^{\frac{3}{5}+\frac{2}{5}}=\\ \boxed{3}$

Therefore, the correct answer is answer a.