To determine the value that remains under the radical when [tex]\(9^{\frac{2}{3}}\)[/tex] is written in its simplest radical form, let's break down the expression step-by-step.
First, recall that 9 can be written as [tex]\(3^2\)[/tex]. So we can rewrite the original expression using this equivalence:
[tex]\[ 9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}} \][/tex]
Next, apply the property of exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (3^2)^{\frac{2}{3}} = 3^{2 \cdot \frac{2}{3}} \][/tex]
Now, multiply the exponents:
[tex]\[ 3^{2 \cdot \frac{2}{3}} = 3^{\frac{4}{3}} \][/tex]
The exponent [tex]\(\frac{4}{3}\)[/tex] can be broken down into:
[tex]\[ 3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} \][/tex]
This can be further separated:
[tex]\[ 3^{1 + \frac{1}{3}} = 3^1 \cdot 3^{\frac{1}{3}} = 3 \cdot 3^{\frac{1}{3}} \][/tex]
Therefore, we see that [tex]\(9^{\frac{2}{3}} = 3 \cdot 3^{\frac{1}{3}}\)[/tex]. The simplest radical form of the expression involves the term [tex]\(3^{\frac{1}{3}}\)[/tex], which means that the number that remains under the radical is 3.
Hence, the correct value that remains under the radical when [tex]\(9^{\frac{2}{3}}\)[/tex] is written in simplest radical form is:
[tex]\[ \boxed{3} \][/tex]
