Question

Which expressions are equivalent to

\[\dfrac{4^{-3}}{4^{-1}}\] ?
Choose 2 answers:
Choose 2 answers:
(Choice A)

\[\dfrac{4^1}{4^3}\]
A

\[\dfrac{4^1}{4^3}\]
(Choice B)

\[\dfrac{1}{4^{2}}\]
B

\[\dfrac{1}{4^{2}}\]
(Choice C)
\[4^3\cdot 4^1\]
C
\[4^3\cdot 4^1\]
(Choice D)
\[(4^{-1})^{-3}\]
D
\[(4^{-1})^{-3}\]

Answer 1

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**.