To solve the problem, we need to determine the value of the expression [tex]\(6^4(6^{-5})\)[/tex] and compare it with the given choices to find all equivalent expressions.
1. Calculate the expression:
The given expression is [tex]\(6^4(6^{-5})\)[/tex]. By the laws of exponents, this can be simplified:
[tex]\[
6^4 \cdot 6^{-5} = 6^{4 + (-5)} = 6^{-1}
\][/tex]
2. Simplify the expression:
The simplified form of [tex]\(6^{-1}\)[/tex] can be rewritten as:
[tex]\[
6^{-1} = \frac{1}{6}
\][/tex]
Now, we will analyze each of the provided choices to see which ones are equivalent to [tex]\(\frac{1}{6}\)[/tex] or [tex]\(6^{-1}\)[/tex]:
- Choice 1: [tex]\(\frac{1}{6^{-1}}\)[/tex]
[tex]\[
\frac{1}{6^{-1}} = \frac{1}{\frac{1}{6}} = 6
\][/tex]
This is not equivalent to [tex]\(6^{-1}\)[/tex] or [tex]\(\frac{1}{6}\)[/tex].
- Choice 2: [tex]\(\frac{1}{6}\)[/tex]
This directly matches [tex]\(\frac{1}{6}\)[/tex].
- Choice 3: [tex]\(6^{-1}\)[/tex]
This is equivalent to [tex]\(\frac{1}{6}\)[/tex].
- Choice 4: [tex]\(\frac{1}{6^1}\)[/tex]
[tex]\[
6^1 = 6 \ \Rightarrow \ \frac{1}{6^1} = \frac{1}{6}
\][/tex]
This is equivalent to [tex]\(\frac{1}{6}\)[/tex].
- Choice 5: [tex]\(\frac{1}{-6}\)[/tex]
[tex]\[
\frac{1}{-6} = -\frac{1}{6}
\][/tex]
This is not equivalent to [tex]\(6^{-1}\)[/tex] or [tex]\(\frac{1}{6}\)[/tex].
Therefore, the choices that are equal to [tex]\(6^4 \left(6^{-5}\right)\)[/tex] and thus [tex]\(\frac{1}{6}\)[/tex] are:
[tex]\[
\boxed{2, 3, 4}
\][/tex]
