To identify which of the given expressions are equivalent to [tex]\(6^{-3}\)[/tex], we first need to evaluate each expression and compare it with [tex]\(6^{-3}\)[/tex].
1. Evaluating [tex]\(6^{-3}\)[/tex]:
[tex]\[
6^{-3} = \frac{1}{6^3}
\][/tex]
To simplify [tex]\(6^{-3}\)[/tex], we need to understand that raising 6 to the power of [tex]\(-3\)[/tex] means taking the reciprocal of [tex]\(6^3\)[/tex].
[tex]\[
6^3 = 6 \times 6 \times 6 = 216
\][/tex]
Therefore,
[tex]\[
6^{-3} = \frac{1}{6^3} = \frac{1}{216}
\][/tex]
2. Evaluating each given expression:
- [tex]\(\frac{1}{6^3}\)[/tex]:
[tex]\[
\frac{1}{6^3} = \frac{1}{216}
\][/tex]
This matches [tex]\(6^{-3}\)[/tex].
- [tex]\(\frac{1}{6^{-3}}\)[/tex]:
Here, we take the reciprocal of [tex]\(6^{-3}\)[/tex].
[tex]\[
\frac{1}{6^{-3}} = 6^3
\][/tex]
Since [tex]\(6^{-3} = \frac{1}{216}\)[/tex], its reciprocal would be:
[tex]\[
6^3 = 216
\][/tex]
This does not match [tex]\(6^{-3}\)[/tex].
- [tex]\(\frac{1}{-216}\)[/tex]:
This expression is simply the reciprocal of [tex]\(-216\)[/tex].
[tex]\[
\frac{1}{-216} = -\frac{1}{216}
\][/tex]
This does not match [tex]\(6^{-3}\)[/tex], as it is the negative of [tex]\(\frac{1}{216}\)[/tex].
- [tex]\(\frac{1}{216}\)[/tex]:
This expression directly evaluates to:
[tex]\[
\frac{1}{216}
\][/tex]
This matches [tex]\(6^{-3}\)[/tex].
3. Conclusion:
From the evaluations above, we can conclude that the expressions equivalent to [tex]\(6^{-3}\)[/tex] are:
[tex]\[
\frac{1}{6^3}
\][/tex]
[tex]\[
\frac{1}{216}
\][/tex]
Thus, the expressions that are equivalent to [tex]\(6^{-3}\)[/tex] are the first and the fourth ones.
So the final answer is:
[tex]\[
\boxed{1 \text{ and } 4}
\][/tex]
