ASSIGNMENT

Which of the following expressions are equivalent to [tex]$6^{-3}$[/tex]?

A. [tex]\frac{1}{6^3}[/tex]
B. [tex]\frac{1}{6^{-3}}[/tex]
C. [tex]\frac{1}{-216}[/tex]
D. [tex]\frac{1}{216}[/tex]



Answer :

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].