To determine which of the given expressions are equivalent to [tex]\(6^{-3}\)[/tex], let's proceed step-by-step.
First, we need to understand what [tex]\(6^{-3}\)[/tex] means:
[tex]\[ 6^{-3} \][/tex]
Using the exponent rule [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex], we can rewrite [tex]\(6^{-3}\)[/tex] as:
[tex]\[ 6^{-3} = \frac{1}{6^3} \][/tex]
Now, let's evaluate the given expressions individually to see which ones match [tex]\( \frac{1}{6^3} \)[/tex].
1. [tex]\(\frac{1}{6^3}\)[/tex]:
[tex]\[
\frac{1}{6^3} = \frac{1}{6^3}
\][/tex]
This is directly equivalent to the rewritten form of [tex]\(6^{-3}\)[/tex].
2. [tex]\(\frac{1}{6^{-3}}\)[/tex]:
[tex]\[
\frac{1}{6^{-3}} = \frac{1}{\frac{1}{6^3}} = 6^3
\][/tex]
This is not equivalent to [tex]\(6^{-3}\)[/tex].
3. [tex]\(\frac{1}{-216}\)[/tex]:
[tex]\[
\frac{1}{-216}
\][/tex]
The value [tex]\(-216\)[/tex] is not related to [tex]\(6^3\)[/tex] (which is [tex]\(216\)[/tex]), so this expression is not equivalent to [tex]\(6^{-3}\)[/tex].
4. [tex]\(\frac{1}{216}\)[/tex]:
[tex]\[
6^3 = 216 \implies \frac{1}{6^3} = \frac{1}{216}
\][/tex]
This is equivalent to the rewritten form of [tex]\(6^{-3}\)[/tex].
So, the equivalent expressions to [tex]\(6^{-3}\)[/tex] are:
- [tex]\(\frac{1}{6^3}\)[/tex]
- [tex]\(\frac{1}{216}\)[/tex]
Therefore, the expressions equivalent to [tex]\(6^{-3}\)[/tex] are: [tex]\(\frac{1}{6^3}\)[/tex] and [tex]\(\frac{1}{216}\)[/tex].
