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:

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

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 separately perform the addition of fractions in the exponent of the expression. 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.