First, we need to simplify the expression [tex]\(5^3 \cdot 5^{-7}\)[/tex].
1. The properties of exponents tell us that when we multiply powers with the same base, we add the exponents. Therefore:
[tex]\[
5^3 \cdot 5^{-7} = 5^{3 + (-7)} = 5^{-4}
\][/tex]
We need to determine which of the given choices are equal to [tex]\(5^{-4}\)[/tex]. Let’s evaluate each option provided:
1. [tex]\(\frac{1}{5^{-4}}\)[/tex]
- By the definition of negative exponents, we know that:
[tex]\[
\frac{1}{5^{-4}} = 5^4
\][/tex]
- This is not equivalent to [tex]\(5^{-4}\)[/tex].
2. [tex]\(\frac{1}{5^4}\)[/tex]
- Simplifying this:
[tex]\[
\frac{1}{5^4}
\][/tex]
- This can be written as:
[tex]\[
5^{-4}
\][/tex]
- This is equivalent to [tex]\(5^{-4}\)[/tex].
3. [tex]\(5^{-4}\)[/tex]
- This is exactly our simplified form. So it is clearly:
[tex]\[
5^{-4}
\][/tex]
- This is equivalent to [tex]\(5^{-4}\)[/tex].
4. [tex]\(5^4\)[/tex]
- This is simply:
[tex]\[
5^4
\][/tex]
- This is not equivalent to [tex]\(5^{-4}\)[/tex].
5. [tex]\(\frac{1}{-625}\)[/tex]
- Evaluating this:
[tex]\[
\frac{1}{-625}
\][/tex]
- Since [tex]\(5^4 = 625\)[/tex], this can be written as:
[tex]\[
\frac{1}{-625} \neq 5^{-4} \quad \text{(because \(-625 < 0\))}
\][/tex]
Thus, the options that are equal to [tex]\(5^3 \cdot 5^{-7}\)[/tex] or [tex]\(5^{-4}\)[/tex] are:
[tex]\[
\boxed{\frac{1}{5^4} \text{ and } 5^{-4}}
\][/tex]
So, the correct selections are the second and third options.
