To express \(\sqrt[5]{3}\) in terms of rational exponents, we need to rewrite the expression using the rules of exponents. The expression \(\sqrt[5]{3}\) can be interpreted as:
\[ 3^{1/5} \]
Next, we apply the square root to this expression. The square root can also be expressed as a fractional exponent:
\[ \sqrt{x} = x^{1/2} \]
Thus, applying the square root to \(3^{1/5}\) gives:
\[ \sqrt[5]{3} = (3^{1/5})^{1/2} \]
Using the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\), we can multiply the exponents:
\[ (3^{1/5})^{1/2} = 3^{(1/5) \cdot (1/2)} = 3^{1/10} \]
Now, we have expressed \(\sqrt[5]{3}\) in terms of a rational exponent:
\[ \sqrt[5]{3} = 3^{1/10} \]
To present this in factored form as requested, we write:
\[ 3^{1/10} = 3^{1} \cdot 3^{-9/10} \text{ (factoring out } 3^{1/10}) \]
This is a valid factorization that shows the rational exponent clearly.
So the answer in factored form for \(\sqrt[5]{3}\) is:
\[ 3^{1/10} \]
(Without explicit factoring but maintaining the expression in rational exponent format.)