To evaluate the numerical expression [tex]\(\left(3^2\right)^{\frac{1}{3}}\)[/tex], let's break it down into clear steps:
1. Calculate [tex]\(3^2\)[/tex]:
The base is 3 and the exponent is 2.
[tex]\[
3^2 = 3 \times 3 = 9
\][/tex]
2. Evaluate [tex]\(\left(9\right)^{\frac{1}{3}}\)[/tex]:
Now we need to find the cube root of 9 because raising a number to the power of [tex]\(\frac{1}{3}\)[/tex] is the same as taking the cube root.
[tex]\[
\left(9\right)^{\frac{1}{3}} \approx 2.080083823051904
\][/tex]
Comparing this step-by-step solution with the multiple-choice options provided:
1. Option 1: 3
- This is incorrect. The expression [tex]\(\left(3^2\right)^{\frac{1}{3}}\)[/tex] does not equal 3.
2. Option 2: [tex]\(\sqrt[3]{9}\)[/tex]
- This is correct. The cube root of 9 is approximately 2.08.
3. Option 3: [tex]\(\sqrt[3]{128}\)[/tex]
- This is incorrect. The cube root of 128 is a different number altogether.
4. Option 4: [tex]\(\sqrt{9}\)[/tex]
- This is incorrect. The square root of 9 is 3, which is not the same as the cube root of [tex]\(3^2\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\sqrt[3]{9}}
\][/tex]
