To simplify the expression [tex]\(\sqrt[3]{v} \cdot \sqrt[5]{v^2}\)[/tex], we proceed as follows:
1. Convert the radicals to exponents:
[tex]\(\sqrt[3]{v}\)[/tex] can be written as [tex]\(v^{1/3}\)[/tex].
[tex]\(\sqrt[5]{v^2}\)[/tex] can be written as [tex]\(v^{2/5}\)[/tex].
2. Multiply the exponents:
When multiplying expressions with the same base, we add the exponents:
[tex]\[
v^{1/3} \cdot v^{2/5} = v^{1/3 + 2/5}
\][/tex]
3. Find a common denominator and add the exponents:
The common denominator for 3 and 5 is 15.
[tex]\[
\frac{1}{3} = \frac{5}{15}
\][/tex]
[tex]\[
\frac{2}{5} = \frac{6}{15}
\][/tex]
Adding these fractions gives:
[tex]\[
\frac{5}{15} + \frac{6}{15} = \frac{11}{15}
\][/tex]
4. Express the product in exponential form:
[tex]\[
v^{1/3 + 2/5} = v^{11/15}
\][/tex]
Therefore, [tex]\(\sqrt[3]{v} \cdot \sqrt[5]{v^2}\)[/tex] simplifies to [tex]\(v^{11/15}\)[/tex].
The simplified form in terms of a radical with at most one radical is:
[tex]\[
v^{11/15} \text{ (as an exponent, this is already in its simplest form, and there is no need to revert back to a radical form as it would complicate the expression unnecessarily)}
\][/tex]
