To simplify the expression [tex]\( 3^{\frac{11}{5}} \div 3^{-\frac{9}{5}} \)[/tex], we can use the properties of exponents. Specifically, we will use the property that states:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
In this case, our base [tex]\(a\)[/tex] is [tex]\(3\)[/tex], [tex]\(m\)[/tex] is [tex]\(\frac{11}{5}\)[/tex], and [tex]\(n\)[/tex] is [tex]\(-\frac{9}{5}\)[/tex]. Applying the property, we get:
[tex]\[
3^{\frac{11}{5}} \div 3^{-\frac{9}{5}} = 3^{\frac{11}{5} - (-\frac{9}{5})}
\][/tex]
Simplify the exponent:
[tex]\[
\frac{11}{5} - (-\frac{9}{5}) = \frac{11}{5} + \frac{9}{5}
\][/tex]
Since the denominators are the same, just add the numerators:
[tex]\[
\frac{11}{5} + \frac{9}{5} = \frac{11 + 9}{5} = \frac{20}{5} = 4
\][/tex]
So the expression simplifies to:
[tex]\[
3^4
\][/tex]
Now, calculate [tex]\(3^4\)[/tex]:
[tex]\[
3^4 = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
Therefore, the simplified value of [tex]\( 3^{\frac{11}{5}} \div 3^{-\frac{9}{5}} \)[/tex] is [tex]\(81\)[/tex].
The correct answer is [tex]\( \boxed{81} \)[/tex].
