Question

Select all the expressions that are equivalent to 8^-5/2^-5
A.) (4^-1)^5
B.) 4
C.) 1/4^-5
D.) 4^-5

Answer 1

To simplify the expression \( \frac{8^{-5}}{2^{-5}} \), we start by rewriting \(8\) in terms of \(2\):

\[
8 = 2^3
\]

Thus:

\[
8^{-5} = (2^3)^{-5} = 2^{-15}
\]

Now, we can rewrite the original expression:

\[
\frac{8^{-5}}{2^{-5}} = \frac{2^{-15}}{2^{-5}} = 2^{-15 - (-5)} = 2^{-15 + 5} = 2^{-10}
\]

Now, we need to evaluate \(2^{-10}\) to check for equivalence with the provided options.

### Evaluating Each Option:

#### A) \( (4^{-1})^5 \)

First, we can simplify \( 4^{-1} \):

\[
4^{-1} = (2^2)^{-1} = 2^{-2}
\]
Now raise to the 5th power:

\[
(4^{-1})^5 = (2^{-2})^5 = 2^{-10}
\]

This option is equivalent to \(2^{-10}\).

#### B) \( 4 \)

We need to write \(4\) as a power of \(2\):

\[
4 = 2^2
\]

This is not equivalent to \(2^{-10}\).

#### C) \( \frac{1}{4^{-5}} \)

First, simplify \(4^{-5}\):

\[
4^{-5} = (2^2)^{-5} = 2^{-10}
\]
Now, take the reciprocal:

\[
\frac{1}{4^{-5}} = \frac{1}{2^{-10}} = 2^{10}
\]

This is not equivalent to \(2^{-10}\).

#### D) \( 4^{-5} \)

Again, simplify \(4^{-5}\):

\[
4^{-5} = (2^2)^{-5} = 2^{-10}
\]

This option is equivalent to \(2^{-10}\).

### Conclusion:

The expressions equivalent to \( \frac{8^{-5}}{2^{-5}} \equiv 2^{-10} \) are:

- A) \( (4^{-1})^5 \)
- D) \( 4^{-5} \)

Thus, the final answer is **A and D**.