To simplify the expression [tex]\( 4^{-\frac{11}{3}} \div 4^{-\frac{2}{3}} \)[/tex], we can use the properties of exponents.
Given the base is the same (4 in this case) and we are dividing the exponents, we can use the rule:
[tex]\[ a^m \div a^n = a^{m-n} \][/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( m = -\frac{11}{3} \)[/tex], and [tex]\( n = -\frac{2}{3} \)[/tex].
So, we subtract the exponents:
[tex]\[ -\frac{11}{3} - (-\frac{2}{3}) \][/tex]
This simplifies to:
[tex]\[ -\frac{11}{3} + \frac{2}{3} \][/tex]
Finding a common denominator, we get:
[tex]\[ -\frac{11}{3} + \frac{2}{3} = -\frac{11 - 2}{3} = -\frac{9}{3} = -3 \][/tex]
So the exponent simplifies to:
[tex]\[ 4^{-3} \][/tex]
Now, we calculate [tex]\( 4^{-3} \)[/tex]. The negative exponent indicates that we can take the reciprocal:
[tex]\[ 4^{-3} = \frac{1}{4^3} \][/tex]
Calculating [tex]\( 4^3 \)[/tex]:
[tex]\[ 4^3 = 64 \][/tex]
So, we get:
[tex]\[ \frac{1}{4^3} = \frac{1}{64} \][/tex]
Therefore, the simplified form of the given expression is [tex]\(\frac{1}{64}\)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{1}{64}} \][/tex]
