To simplify the given expression:
[tex]\[
4^{-\frac{11}{3}} \div 4^{-\frac{2}{3}}
\][/tex]
we will use properties of exponents. Specifically, we use the rule that states:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
Here, [tex]\(a = 4\)[/tex], [tex]\(m = -\frac{11}{3}\)[/tex], and [tex]\(n = -\frac{2}{3}\)[/tex]. Substituting these values into the rule, we get:
[tex]\[
4^{-\frac{11}{3}} \div 4^{-\frac{2}{3}} = 4^{-\frac{11}{3} - (-\frac{2}{3})}
\][/tex]
Next, we need to perform the subtraction inside the exponent:
[tex]\[
-\frac{11}{3} - (-\frac{2}{3}) = -\frac{11}{3} + \frac{2}{3}
\][/tex]
To subtract these fractions, we combine them into a single fraction:
[tex]\[
-\frac{11}{3} + \frac{2}{3} = -\frac{11 - 2}{3} = -\frac{9}{3}
\][/tex]
Now, simplify the fraction:
[tex]\[
-\frac{9}{3} = -3
\][/tex]
So, the expression simplifies to:
[tex]\[
4^{-3}
\][/tex]
To further simplify, we use the property of negative exponents, which states:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Applying this property to our expression:
[tex]\[
4^{-3} = \frac{1}{4^3}
\][/tex]
Now calculate [tex]\(4^3\)[/tex]:
[tex]\[
4^3 = 4 \times 4 \times 4 = 64
\][/tex]
Therefore:
[tex]\[
4^{-3} = \frac{1}{64}
\][/tex]
Hence, the simplified form of the expression is:
[tex]\[
\frac{1}{64}
\][/tex]
The correct answer is:
A. [tex]\(\frac{1}{64}\)[/tex]
