To determine which expressions are equivalent to [tex]\(\frac{1}{36}\)[/tex], we will evaluate each expression step-by-step and then compare the results.
1. Evaluate [tex]\(3^{-6}\)[/tex]:
[tex]\[
3^{-6} = \frac{1}{3^6} \approx 0.0013717
\][/tex]
This is not equal to [tex]\(\frac{1}{36}\)[/tex].
2. Evaluate [tex]\(6^{-2}\)[/tex]:
[tex]\[
6^{-2} = \frac{1}{6^2} = \frac{1}{36} = 0.027777777777777776
\][/tex]
This is equal to [tex]\(\frac{1}{36}\)[/tex].
3. Evaluate [tex]\(\frac{6^3}{6^5}\)[/tex]:
Using the properties of exponents, we can write it as:
[tex]\[
\frac{6^3}{6^5} = 6^{3-5} = 6^{-2} = \frac{1}{6^2} = \frac{1}{36} = 0.027777777777777776
\][/tex]
This is equal to [tex]\(\frac{1}{36}\)[/tex].
4. Evaluate [tex]\(\frac{6^2}{6^{-1}}\)[/tex]:
Using the properties of exponents, we can write it as:
[tex]\[
\frac{6^2}{6^{-1}} = 6^{2-(-1)} = 6^{2+1} = 6^3 = 216
\][/tex]
This is not equal to [tex]\(\frac{1}{36}\)[/tex].
5. Evaluate [tex]\(6^{-2}\)[/tex] (same as the 2nd expression):
[tex]\[
6^{-2} = \frac{1}{6^2} = \frac{1}{36} = 0.027777777777777776
\][/tex]
This is equal to [tex]\(\frac{1}{36}\)[/tex].
6. Evaluate [tex]\(6^{-9} \cdot 8^7\)[/tex]:
[tex]\[
6^{-9} \cdot 8^7 \approx 0.208098
\][/tex]
This is not equal to [tex]\(\frac{1}{36}\)[/tex].
From the evaluations, the expressions that are equivalent to [tex]\(\frac{1}{36}\)[/tex] are:
[tex]\[
\boxed{6^{-2}, \frac{6^3}{6^5}, \text{ and } 6^{-2}}
\][/tex]
