Certainly! Let's solve the expression [tex]\( 9^{\frac{2}{3}} \)[/tex] and simplify it to find out which value remains under the radical.
1. Express [tex]\( 9 \)[/tex] in terms of its prime factors:
The number [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex]. Thus, the expression [tex]\( 9^{\frac{2}{3}} \)[/tex] can be rewritten using its prime factors:
[tex]\[
9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}}
\][/tex]
2. Simplify the exponent:
Using the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify [tex]\((3^2)^{\frac{2}{3}}\)[/tex]:
[tex]\[
(3^2)^{\frac{2}{3}} = 3^{2 \cdot \frac{2}{3}}
\][/tex]
3. Multiply the exponents:
Next, we multiply [tex]\( 2 \)[/tex] and [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[
3^{2 \cdot \frac{2}{3}} = 3^{\frac{4}{3}}
\][/tex]
4. Express in simplest radical form:
The exponent [tex]\( \frac{4}{3} \)[/tex] indicates that the base [tex]\( 3 \)[/tex] is raised to the fourth power and then the cube root is taken. This can be written as:
[tex]\[
3^{\frac{4}{3}} = \sqrt[3]{3^4}
\][/tex]
Since [tex]\( 3^4 = 81 \)[/tex], we have:
[tex]\[
\sqrt[3]{3^4} = \sqrt[3]{81}
\][/tex]
So, in the given notation [tex]\( 9^{\frac{2}{3}} \)[/tex], when simplified to its radical form, we get [tex]\( \sqrt[3]{81} \)[/tex].
The value that remains under the radical is the base value of [tex]\( 3 \)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{3}
\][/tex]
