To simplify the expression [tex]\( 3^{\frac{11}{5}} \div 3^{-\frac{9}{5}} \)[/tex], follow these steps:
1. Review Properties of Exponents:
Remember the property of exponents: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]. This property will help simplify the division of two exponents with the same base.
2. Combine Exponents Using the Property:
Here, the expression can be rewritten with the exponents subtracted:
[tex]\[
3^{\frac{11}{5}} \div 3^{-\frac{9}{5}} = 3^{\left(\frac{11}{5} - \left(-\frac{9}{5}\right)\right)}
\][/tex]
Simplify the exponent:
[tex]\[
\frac{11}{5} - \left(-\frac{9}{5}\right) = \frac{11}{5} + \frac{9}{5}
\][/tex]
Add the exponents:
[tex]\[
\frac{11}{5} + \frac{9}{5} = \frac{20}{5} = 4
\][/tex]
3. Simplify the Base:
Now the expression simplifies to:
[tex]\[
3^4
\][/tex]
4. Calculate the Final Value:
Finally, calculate [tex]\(3^4\)[/tex]:
[tex]\[
3^4 = 81
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{81}
\][/tex]
