Let's solve the problem step-by-step.
We need to evaluate the expression [tex]\(\sqrt[3]{\left(-\frac{2}{3}\right)^9}\)[/tex].
1. Determine the base and the exponent:
The base is [tex]\(-\frac{2}{3}\)[/tex], and the exponent is 9.
2. Raise the base to the exponent:
Calculate [tex]\(\left(-\frac{2}{3}\right)^9\)[/tex]:
[tex]\[
\left(-\frac{2}{3}\right)^9 = -0.02601229487374891
\][/tex]
3. Take the cube root of the result:
We need to evaluate [tex]\(\sqrt[3]{-0.02601229487374891}\)[/tex]:
[tex]\[
\sqrt[3]{-0.02601229487374891} = 0.14814814814814817 + 0.25660011963983365i
\][/tex]
So, through these steps, we find:
[tex]\[
\sqrt[3]{\left(-\frac{2}{3}\right)^9} = 0.14814814814814817 + 0.25660011963983365i
\][/tex]
This result indicates that the cube root of [tex]\((-2/3)^9\)[/tex] yields a complex number. This is the final answer.
