To simplify the expression [tex]\(\left(x^{\frac{1}{6}}\right)^3\)[/tex], we can use the rules of exponents. Here's the step-by-step process:
1. Exponentiation Rule: When you raise a power to a power, you multiply the exponents. For any base [tex]\(a\)[/tex] and exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex], the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] applies.
2. Application to the Problem: Applying this rule to our expression:
[tex]\[
\left(x^{\frac{1}{6}}\right)^3 = x^{\frac{1}{6} \cdot 3}
\][/tex]
3. Multiplication of Exponents: Multiply [tex]\(\frac{1}{6}\)[/tex] by 3:
[tex]\[
\frac{1}{6} \cdot 3 = \frac{3}{6} = \frac{1}{2}
\][/tex]
4. Result: This simplifies the expression to:
[tex]\[
x^{\frac{1}{2}}
\][/tex]
So, the simplified form of [tex]\(\left(x^{\frac{1}{6}}\right)^3\)[/tex] is [tex]\(x^{\frac{1}{2}}\)[/tex].
Among the given options:
- [tex]\(x^2\)[/tex]
- [tex]\(x^3\)[/tex]
- [tex]\(x^{\frac{1}{3}}\)[/tex]
- [tex]\(x^{\frac{1}{2}}\)[/tex]
The correct simplified form is:
[tex]\[
\boxed{x^{\frac{1}{2}}}
\][/tex]
