Sure! Let's simplify the given expression step by step.
We start with the expression:
[tex]\[
3^{\frac{11}{5}} \div 3^{-\frac{2}{5}}
\][/tex]
Using the rule of exponents which states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can rewrite the division of exponents as a single exponent with the base:
[tex]\[
3^{\frac{11}{5}} \div 3^{-\frac{2}{5}} = 3^{\left(\frac{11}{5} - (-\frac{2}{5})\right)}
\][/tex]
Simplifying the exponent:
[tex]\[
\frac{11}{5} - (-\frac{2}{5}) = \frac{11}{5} + \frac{2}{5} = \frac{11 + 2}{5} = \frac{13}{5}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
3^{\frac{13}{5}}
\][/tex]
Now, let's determine which of the provided options corresponds to this exponentiation. Simplifying the numerical value of [tex]\( 3^{\frac{13}{5}} \)[/tex]:
[tex]\[
3^{\frac{13}{5}} \approx 17.398638404385867
\][/tex]
Looking at the given answer choices:
A. 12
B. 81
C. [tex]\(\frac{1}{81}\)[/tex]
D. [tex]\(\frac{1}{12}\)[/tex]
None of these choices perfectly match the exact value. Since [tex]\(17.398638404385867\)[/tex] is the value obtained by raising 3 to the power of [tex]\(\frac{13}{5}\)[/tex], we can see that none of the provided options correctly represents this value.
Therefore, none of the given answer choices (A, B, C, or D) are correct based on the rules and simplifications we've applied.
