To simplify the expression \( 4^{-\frac{11}{3}} \div 4^{-\frac{2}{3}} \), we will use the properties of exponents. Specifically, we use the property that when dividing two expressions with the same base, we subtract the exponents:
[tex]\[
a^m \div a^n = a^{m-n}
\][/tex]
Given:
[tex]\[
4^{-\frac{11}{3}} \div 4^{-\frac{2}{3}}
\][/tex]
We set \( a = 4 \), \( m = -\frac{11}{3} \), and \( n = -\frac{2}{3} \). Applying the property of exponents, we get:
[tex]\[
4^{-\frac{11}{3}} \div 4^{-\frac{2}{3}} = 4^{-\frac{11}{3} - (-\frac{2}{3})}
\][/tex]
To simplify, subtracting the exponents:
[tex]\[
-\frac{11}{3} - (-\frac{2}{3}) = -\frac{11}{3} + \frac{2}{3}
\][/tex]
Combine the fractions:
[tex]\[
-\frac{11}{3} + \frac{2}{3} = -\frac{11}{3} + \frac{2}{3} = -\frac{11 - 2}{3} = -\frac{9}{3} = -3
\][/tex]
Thus, the expression simplifies to:
[tex]\[
4^{-\frac{9}{3}} = 4^{-3}
\][/tex]
To simplify \( 4^{-3} \):
[tex]\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{1}{64}}
\][/tex]
