To simplify the expression [tex]\(x^{\frac{1}{3}} \cdot x^{\frac{1}{5}}\)[/tex], we need to use the properties of exponents. Specifically, when multiplying terms with the same base, we add the exponents. Here’s a step-by-step solution:
1. Identify the exponents in the given expression. The exponents are [tex]\( \frac{1}{3} \)[/tex] and [tex]\( \frac{1}{5} \)[/tex].
2. Add the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{5}
\][/tex]
3. To add these fractions, we need a common denominator. The least common denominator (LCD) of 3 and 5 is 15.
4. Convert each fraction to an equivalent fraction with a denominator of 15:
[tex]\[
\frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{1}{5} = \frac{3}{15}
\][/tex]
5. Add the fractions:
[tex]\[
\frac{5}{15} + \frac{3}{15} = \frac{8}{15}
\][/tex]
6. Substitute the combined exponent back into the expression:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{5}} = x^{\frac{8}{15}}
\][/tex]
So, the simplified expression is [tex]\( x^{\frac{8}{15}} \)[/tex].
The correct answer is:
A. [tex]\( x^{\frac{8}{15}} \)[/tex]
