Certainly! 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]
Recall the property of exponents that states:
[tex]\[
a^m \div a^n = a^{m-n}
\][/tex]
Using this property, we can combine the two powers of 3 by subtracting the exponents:
[tex]\[
3^{\frac{11}{5}} \div 3^{-\frac{2}{5}} = 3^{\left(\frac{11}{5} - \left(-\frac{2}{5}\right)\right)}
\][/tex]
Next, simplify the exponent:
[tex]\[
\frac{11}{5} - \left(-\frac{2}{5}\right) = \frac{11}{5} + \frac{2}{5}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{11}{5} + \frac{2}{5} = \frac{11 + 2}{5} = \frac{13}{5}
\][/tex]
So, the expression simplifies to:
[tex]\[
3^{\frac{13}{5}}
\][/tex]
To further simplify, we can convert [tex]\(\frac{13}{5}\)[/tex] to a decimal. This gives us:
[tex]\[
3^{2.6}
\][/tex]
We need to find the numerical value of [tex]\(3^{2.6}\)[/tex]. Calculating this value, we get approximately:
[tex]\[
3^{2.6} \approx 17.398638404385867
\][/tex]
This value does not directly match any of the provided options (A. [tex]\(\frac{1}{81}\)[/tex], B. [tex]\(\frac{1}{12}\)[/tex], C. 81, D. 12)).
Upon re-examining the given options, none of them approximate [tex]\(3^{2.6}\)[/tex] closely. Therefore, the correct simplification of the expression is:
[tex]\[
3^{2.6} \approx 17.398638404385867
\][/tex]
None of the provided multiple-choice answers (A, B, C, D) are correct based on our simplification and calculation.
