To find which expressions are equivalent to [tex]\(\frac{3^{-8}}{3^{-4}}\)[/tex], we can start by simplifying the exponent within the given expression.
### Step-by-Step Simplification:
1. Simplify the Exponents:
[tex]\[
\frac{3^{-8}}{3^{-4}} = 3^{-8 - (-4)}
\][/tex]
2. Combine the Exponents:
Since subtracting a negative is the same as adding the positive:
[tex]\[
\frac{3^{-8}}{3^{-4}} = 3^{-8 + 4} = 3^{-4}
\][/tex]
So, [tex]\(\frac{3^{-8}}{3^{-4}} = 3^{-4}\)[/tex].
### Equivalent Expressions:
Now we need to find which of the given options are equivalent to [tex]\(3^{-4}\)[/tex].
#### Exploring the Options:
- Option A: [tex]\(3^{-12}\)[/tex]
[tex]\[
3^{-12} \neq 3^{-4}
\][/tex]
This option is not equivalent.
- Option B: [tex]\(3^{-4}\)[/tex]
[tex]\[
3^{-4} = 3^{-4}
\][/tex]
This option is equivalent.
- Option C: [tex]\(3^2\)[/tex]
[tex]\[
3^2 \neq 3^{-4}
\][/tex]
This option is not equivalent.
- Option D: [tex]\(\frac{1}{3^2}\)[/tex]
[tex]\[
\frac{1}{3^2} = 3^{-2} \neq 3^{-4}
\][/tex]
This option is not equivalent.
- Option E: [tex]\(\frac{1}{3^4}\)[/tex]
[tex]\[
\frac{1}{3^4} = 3^{-4}
\][/tex]
This option is equivalent.
- Option F: [tex]\(\frac{1}{3^{12}}\)[/tex]
[tex]\[
\frac{1}{3^{12}} = 3^{-12} \neq 3^{-4}
\][/tex]
This option is not equivalent.
### Conclusion:
The expressions equivalent to [tex]\(\frac{3^{-8}}{3^{-4}}\)[/tex] are:
- B: [tex]\(3^{-4}\)[/tex]
- E: [tex]\(\frac{1}{3^4}\)[/tex]
So, the correct selections are [tex]\(\boxed{B \text{ and } E}\)[/tex].
