To simplify the expression [tex]\( x^{\frac{1}{3}} \cdot x^{\frac{1}{6}} \)[/tex], we can use the property of exponents that states [tex]\( x^a \cdot x^b = x^{a+b} \)[/tex].
1. Identify the exponents in the given expression:
[tex]\[
x^{\frac{1}{3}} \quad \text{and} \quad x^{\frac{1}{6}}
\][/tex]
2. Add the exponents:
[tex]\[
\frac{1}{3} + \frac{1}{6}
\][/tex]
3. To add these fractions, find a common denominator. The least common denominator of 3 and 6 is 6.
4. Convert [tex]\(\frac{1}{3}\)[/tex] to a fraction with a denominator of 6:
[tex]\[
\frac{1}{3} = \frac{2}{6}
\][/tex]
5. Now add the fractions:
[tex]\[
\frac{2}{6} + \frac{1}{6} = \frac{3}{6}
\][/tex]
6. Simplify [tex]\(\frac{3}{6}\)[/tex]:
[tex]\[
\frac{3}{6} = \frac{1}{2}
\][/tex]
So, the simplified expression is:
[tex]\[
x^{\frac{1}{2}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x^{\frac{1}{2}}}
\][/tex]
Given the provided options, none of them match exactly with [tex]\( x^{\frac{1}{2}} \)[/tex], but the closest one that represents the simplified form is not listed. The closest approximation of the correct exponent based on the fractions is not provided, so it looks like none of the given options (A, B, C, or D) are correct.
