To find which value remains under the radical when [tex]\( 9^{\frac{2}{3}} \)[/tex] is simplified, let's break it down step-by-step.
1. Expressing 9 in terms of its prime factor:
[tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex], so:
[tex]\[
9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}}
\][/tex]
2. Applying the power of a power property:
When you raise a power to another power, you multiply the exponents:
[tex]\[
(3^2)^{\frac{2}{3}} = 3^{2 \cdot \frac{2}{3}} = 3^{\frac{4}{3}}
\][/tex]
3. Simplifying the exponent:
The exponent [tex]\( \frac{4}{3} \)[/tex] can be decomposed into:
[tex]\[
3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}}
\][/tex]
This can be further split into:
[tex]\[
3^{1 + \frac{1}{3}} = 3^1 \cdot 3^{\frac{1}{3}}
\][/tex]
Which simplifies to:
[tex]\[
3 \cdot 3^{\frac{1}{3}}
\][/tex]
4. Identifying the value under the radical:
From the expression above, [tex]\( 3^{\frac{1}{3}} \)[/tex] represents the cube root of 3.
Therefore, the value that remains under the radical when [tex]\( 9^{\frac{2}{3}} \)[/tex] is written in simplest radical form is [tex]\( 3 \)[/tex].
So, the correct answer is:
3
