To determine which expressions are equivalent to [tex]\(\frac{4^{-3}}{4^{-1}}\)[/tex], we will simplify this expression and compare it with the given options.
First, let's simplify the given expression [tex]\(\frac{4^{-3}}{4^{-1}}\)[/tex] step by step:
[tex]\[
\frac{4^{-3}}{4^{-1}}
\][/tex]
Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{4^{-3}}{4^{-1}} = 4^{-3 - (-1)} = 4^{-3 + 1} = 4^{-2}
\][/tex]
So, [tex]\(\frac{4^{-3}}{4^{-1}} = 4^{-2}\)[/tex].
Next, we will check each given option to see which ones simplify to [tex]\(4^{-2}\)[/tex].
(A) [tex]\(\frac{4^1}{4^3}\)[/tex]
[tex]\[
\frac{4^1}{4^3} = 4^{1-3} = 4^{-2}
\][/tex]
This is equivalent to [tex]\(4^{-2}\)[/tex].
(B) [tex]\(\frac{1}{4^2}\)[/tex]
[tex]\[
\frac{1}{4^2} = 4^{-2}
\][/tex]
This is equivalent to [tex]\(4^{-2}\)[/tex].
(C) [tex]\(4^3 \cdot 4^1\)[/tex]
[tex]\[
4^3 \cdot 4^1 = 4^{3+1} = 4^4
\][/tex]
This is not equivalent to [tex]\(4^{-2}\)[/tex].
(D) [tex]\((4^{-1})^{-3}\)[/tex]
[tex]\[
(4^{-1})^{-3} = 4^{-1 \cdot -3} = 4^3
\][/tex]
This is not equivalent to [tex]\(4^{-2}\)[/tex].
Thus, the expressions that are equivalent to [tex]\(\frac{4^{-3}}{4^{-1}}\)[/tex] are:
- (A) [tex]\(\frac{4^1}{4^3}\)[/tex]
- (B) [tex]\(\frac{1}{4^2}\)[/tex]
