To solve this problem, we need to find which two expressions from the given list are equivalent to [tex]\( 4^{-3} \)[/tex].
First, recall the definition and properties of negative exponents:
[tex]\[
4^{-3} = \frac{1}{4^3}
\][/tex]
Now, let's analyze each of the given expressions step-by-step:
1. [tex]\(4 \times 4 \times 4\)[/tex]:
[tex]\[
4 \times 4 \times 4 = 4^3
\][/tex]
This is not equivalent to [tex]\(4^{-3}\)[/tex].
2. [tex]\(\frac{1}{4^3}\)[/tex]:
[tex]\[
\frac{1}{4^3} = 4^{-3}
\][/tex]
This is equivalent to [tex]\(4^{-3}\)[/tex].
3. [tex]\(\frac{1}{4 \times 4 \times 4}\)[/tex]:
[tex]\[
\frac{1}{4 \times 4 \times 4} = \frac{1}{4^3} = 4^{-3}
\][/tex]
This is equivalent to [tex]\(4^{-3}\)[/tex].
4. [tex]\(\frac{1}{3^4}\)[/tex]:
[tex]\[
\frac{1}{3^4} \neq 4^{-3}
\][/tex]
This is not equivalent to [tex]\(4^{-3}\)[/tex].
5. [tex]\(\frac{1}{3 \times 3 \times 3 \times 3}\)[/tex]:
[tex]\[
\frac{1}{3 \times 3 \times 3 \times 3} = \frac{1}{3^4} \neq 4^{-3}
\][/tex]
This is not equivalent to [tex]\(4^{-3}\)[/tex].
6. [tex]\(\frac{1}{4 \times 3}\)[/tex]:
[tex]\[
\frac{1}{4 \times 3} = \frac{1}{12} \neq 4^{-3}
\][/tex]
This is not equivalent to [tex]\(4^{-3}\)[/tex].
Comparing all the expressions, we find that the two expressions that are equivalent to [tex]\( 4^{-3} \)[/tex] are:
- [tex]\(\frac{1}{4^3}\)[/tex]
- [tex]\(\frac{1}{4 \times 4 \times 4}\)[/tex]
Thus, the expressions that are equivalent to [tex]\( 4^{-3} \)[/tex] are the second and third ones.
