Answer :
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]
[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]
