To simplify the expression [tex]\(x^{\frac{1}{3}} \cdot x^{\frac{1}{5}}\)[/tex], we can use the properties of exponents. Specifically, we use the property that when multiplying two exponents with the same base, we add the exponents together. Here's a step-by-step explanation:
1. Identify the exponents: We are multiplying [tex]\(x\)[/tex] raised to [tex]\(\frac{1}{3}\)[/tex] and [tex]\(x\)[/tex] raised to [tex]\(\frac{1}{5}\)[/tex].
2. Add the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{5}
\][/tex]
3. Find a common denominator to add the fractions:
- The common denominator for 3 and 5 is 15.
4. Convert the fractions:
[tex]\[
\frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{1}{5} = \frac{3}{15}
\][/tex]
5. Add the converted fractions:
[tex]\[
\frac{5}{15} + \frac{3}{15} = \frac{8}{15}
\][/tex]
6. Combine the fractions:
[tex]\[
\frac{1}{3} + \frac{1}{5} = \frac{8}{15}
\][/tex]
7. Simplified expression:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{5}} = x^{\frac{8}{15}}
\][/tex]
Based on the given choices:
A. [tex]\(\quad x^{\frac{1}{16}}\)[/tex]
B. [tex]\(\quad x^{15}\)[/tex]
C. [tex]\(\quad x^{\frac{8}{16}}\)[/tex]
D. [tex]\(\quad x^{\frac{2}{15}}\)[/tex]
The correct answer is:
[tex]\[
\boxed{x^{\frac{8}{15}}}
\][/tex]
