Question

Select all terms that are equivalent to 4^3/4^5


Show your work.

(2 points)
Responses

1/4^2

4^2

4^−2

1/16

−4^2

Answer 1

To simplify the expression \( \frac{4^3}{4^5} \), we can use the properties of exponents. Specifically, when dividing with the same base, we subtract the exponents:

\[
\frac{a^m}{a^n} = a^{m-n}
\]

Applying this to our expression:

\[
\frac{4^3}{4^5} = 4^{3-5} = 4^{-2}
\]

Now, let's consider the equivalent terms.

1. **\( \frac{1}{4^2} \)**: Since \( 4^{-2} = \frac{1}{4^2} \), this is equivalent.

2. **\( 4^2 \)**: This is not equivalent to \( 4^{-2} \).

3. **\( 4^{-2} \)**: This is the simplified form, so it is equivalent.

4. **\( \frac{1}{16} \)**: Since \( 4^2 = 16 \), thus \( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \). This is equivalent.

5. **\( -4^2 \)**: This simplifies to \(-16\), which is not equivalent to \( 4^{-2} \).

Based on the analysis, the terms that are equivalent to \( \frac{4^3}{4^5} \) are:

- \( \frac{1}{4^2} \)
- \( 4^{-2} \)
- \( \frac{1}{16} \)

So, the final equivalent terms are:

- \( \frac{1}{4^2} \)
- \( 4^{-2} \)
- \( \frac{1}{16} \)