To simplify the given expression
[tex]\[ 3^{\frac{11}{8}} \div 3^{-\frac{9}{6}} \][/tex]
we need to handle the division of exponents with the same base using the properties of exponents. The property that applies here is
[tex]\[ \frac{a^m}{a^n} = a^{m - n} \][/tex]
Thus,
[tex]\[ 3^{\frac{11}{8}} \div 3^{-\frac{9}{6}} = 3^{\frac{11}{8} - \left( -\frac{9}{6} \right)} \][/tex]
First, we need to find a common denominator to subtract these fractions. The given exponents are [tex]\(\frac{11}{8}\)[/tex] and [tex]\(\frac{-9}{6}\)[/tex].
To subtract the exponents, we need to convert [tex]\(\frac{-9}{6}\)[/tex] into a fraction with a denominator of 8. Converting [tex]\(\frac{-9}{6}\)[/tex]:
[tex]\[ \frac{-9}{6} = \frac{-9 \times 4}{6 \times 4} = \frac{-36}{24} = \frac{-12}{8} \][/tex]
Now, substitute and subtract these fractions:
[tex]\[ \frac{11}{8} - \left(-\frac{12}{8}\right) = \frac{11}{8} + \frac{12}{8} = \frac{11 + 12}{8} = \frac{23}{8} \][/tex]
So the expression simplifies to:
[tex]\[ 3^{\frac{23}{8}} \][/tex]
We need to calculate [tex]\( 3^{\frac{23}{8}} \)[/tex]. Converting [tex]\(\frac{23}{8}\)[/tex] to a decimal:
[tex]\[ \frac{23}{8} = 2.875 \][/tex]
Now we calculate:
[tex]\[ 3^{2.875} \approx 23.535509657536863 \][/tex]
This result corresponds closely to a known integer power of 3. Through analysis, we know that
[tex]\[ 3^3 = 27 \][/tex]
and since 23.535509657536863 is fairly close to this value, it isn't a convenient integer match but should be compared with the given choices. The closest correct value from the given choices, representing similar magnitude, would be
[tex]\[ \boxed{12} \][/tex]
As per multiple choice method, when considered options are seen, the answer should be identified as
[tex]\[ \boxed{12} \][/tex]
