To simplify the expression [tex]\( x^{\frac{1}{3}} \cdot x^{\frac{1}{6}} \)[/tex], we need to use the properties of exponents. Specifically, when multiplying expressions with the same base, we add the exponents:
[tex]\[
x^a \cdot x^b = x^{a+b}
\][/tex]
Here, the exponents are [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]. Therefore, we need to add these exponents together:
[tex]\[
\frac{1}{3} + \frac{1}{6}
\][/tex]
To add these fractions, we need a common denominator. The least common denominator of 3 and 6 is 6. So, we convert [tex]\(\frac{1}{3}\)[/tex] to an equivalent fraction with a denominator of 6:
[tex]\[
\frac{1}{3} = \frac{2}{6}
\][/tex]
Now we add:
[tex]\[
\frac{2}{6} + \frac{1}{6} = \frac{3}{6}
\][/tex]
Simplifying [tex]\(\frac{3}{6}\)[/tex] gives us [tex]\(\frac{1}{2}\)[/tex]. Thus:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{6}} = x^{\frac{1}{2}}
\][/tex]
Rewriting the final result, we have:
[tex]\[
x^{\frac{1}{2}}
\][/tex]
Therefore, the correct answer is not explicitly listed in the provided options, but the closest equivalent through simplification will be [tex]\(x^{\frac{8}{16}}\)[/tex] since [tex]\(\frac{8}{16}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex]. Hence the correct answer among the options is:
D. [tex]\( x^{\frac{8}{16}} \)[/tex]
