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

Solve the following exercise:

$\sqrt[5]{3^3}\cdot\sqrt[5]{3^2}=$

In order to simplify the given expression, we will apply 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:

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

Begin by converting the roots to exponent notation using the law of exponents mentioned in a:

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

Given that we are multiplying two terms with identical bases, we'll apply the law of exponents mentioned in b:

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

Proceed to perform the addition of fractions in the exponent of the expression separately. We can achieve 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:

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

We obtain the following:

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

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.