To solve the problem of finding an expression equivalent to [tex]\(x^{\frac{1}{6}} \cdot x^{\frac{1}{6}}\)[/tex], we need to use the properties of exponents.
### Step-by-Step Solution
1. Identify the exponents: We have two exponents that are both [tex]\(\frac{1}{6}\)[/tex].
[tex]\[ x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} \][/tex]
2. Use the multiplication property of exponents: According to the properties of exponents, when multiplying terms with the same base, we add their exponents.
[tex]\[ x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} = x^{\frac{1}{6} + \frac{1}{6}} \][/tex]
3. Add the exponents:
[tex]\[ \frac{1}{6} + \frac{1}{6} = \frac{2}{6} \][/tex]
4. Simplify the exponent:
[tex]\[ \frac{2}{6} = \frac{1}{3} \][/tex]
Thus, the expression simplifies to:
[tex]\[ x^{\frac{1}{3}} \][/tex]
5. Match the simplified expression with the given choices:
- [tex]\(\frac{1}{\sqrt[3]{x}}\)[/tex]
- [tex]\(\sqrt{x}\)[/tex]
- [tex]\(\sqrt[3]{x}\)[/tex]
- [tex]\(\sqrt[36]{x}\)[/tex]
The simplified expression [tex]\(x^{\frac{1}{3}}\)[/tex] matches the cube root of [tex]\(x\)[/tex]:
[tex]\[ x^{\frac{1}{3}} = \sqrt[3]{x} \][/tex]
### Conclusion
Therefore, the expression [tex]\(x^{\frac{1}{6}} \cdot x^{\frac{1}{6}}\)[/tex] is equivalent to:
[tex]\(\sqrt[3]{x}\)[/tex]
So, the correct option is:
[tex]\(\sqrt[3]{x}\)[/tex]
