To simplify the expression [tex]\( x^{\frac{1}{3}} \cdot x^{\frac{1}{5}} \)[/tex], we use the property of exponents that states [tex]\( x^a \cdot x^b = x^{a+b} \)[/tex].
Here, the exponents are [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex]. Our task is to add these two fractions:
[tex]\[
\frac{1}{3} + \frac{1}{5}
\][/tex]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 5 is 15, so we convert both fractions to have this common denominator:
[tex]\[
\frac{1}{3} = \frac{5}{15}
\][/tex]
[tex]\[
\frac{1}{5} = \frac{3}{15}
\][/tex]
Next, we add the fractions:
[tex]\[
\frac{5}{15} + \frac{3}{15} = \frac{5 + 3}{15} = \frac{8}{15}
\][/tex]
Therefore, the expression [tex]\( x^{\frac{1}{3}} \cdot x^{\frac{1}{5}} \)[/tex] simplifies to:
[tex]\[
x^{\frac{8}{15}}
\][/tex]
The correct answer is:
A. [tex]\( x^{\frac{8}{15}} \)[/tex]
