To rewrite the expression [tex]\(\left(3^{\frac{2}{3}}\right)^{\frac{1}{6}}\)[/tex] with a rational exponent as a radical expression, we can follow these steps:
1. Start with the given expression:
[tex]\[
\left(3^{\frac{2}{3}}\right)^{\frac{1}{6}}
\][/tex]
2. Use the property of exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Apply this property to combine the exponents:
[tex]\[
3^{\left(\frac{2}{3} \cdot \frac{1}{6}\right)}
\][/tex]
3. Multiply the exponents [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\frac{2}{3} \cdot \frac{1}{6} = \frac{2 \times 1}{3 \times 6} = \frac{2}{18} = \frac{1}{9}
\][/tex]
4. Now the expression is:
[tex]\[
3^{\frac{1}{9}}
\][/tex]
5. The exponent [tex]\(\frac{1}{9}\)[/tex] indicates the 9th root of 3. Therefore, we can rewrite the expression as a radical expression:
[tex]\[
\sqrt[9]{3}
\][/tex]
So, the radical expression equivalent to [tex]\(\left(3^{\frac{2}{3}}\right)^{\frac{1}{6}}\)[/tex] is:
[tex]\[
\sqrt[9]{3}
\][/tex]
