To solve the expression \(\dfrac{4^{-3}}{4^{-1}}\), we can use the quotient of powers property, which states that \(\dfrac{a^m}{a^n} = a^{m-n}\).
Calculating this, we have:
\[ \dfrac{4^{-3}}{4^{-1}} = 4^{-3 - (-1)} = 4^{-3 + 1} = 4^{-2} \]
Now we will evaluate each of the given choices to see if they are equivalent to \(4^{-2}\).
Choice A: \(\dfrac{4^1}{4^3}\)
Using the quotient of powers property:
\[ \dfrac{4^1}{4^3} = 4^{1 - 3} = 4^{-2} \] This is equivalent to \(4^{-2}\).
Choice B: \(\dfrac{1}{4^2}\)
We can rewrite this as:
\[ \dfrac{1}{4^2} = 4^{-2} \] This is also equivalent to \(4^{-2}\).
Choice C: \(4^3 \cdot 4^1\)
Using the product of powers property, we have:
\[ 4^3 \cdot 4^1 = 4^{3 + 1} = 4^4 \] This is not equivalent to \(4^{-2}\).
Choice D: \((4^{-1})^{-3}\)
Using the power of a power property, we calculate:
\[ (4^{-1})^{-3} = 4^{-1 \cdot -3} = 4^{3} \] This is not equivalent to \(4^{-2}\).
Conclusion
The expressions that are equivalent to \(\dfrac{4^{-3}}{4^{-1}}\) are:
- Choice A: \(\dfrac{4^1}{4^3}\)
- Choice B: \(\dfrac{1}{4^2}\)
So, the two answers are A and B.