To determine which of the given expressions are equal to [tex]\(5^3 \cdot 5^{-7}\)[/tex], we need to simplify this expression step-by-step and compare it with each option.
### Simplifying [tex]\(5^3 \cdot 5^{-7}\)[/tex]:
Using the properties of exponents, specifically that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can simplify [tex]\(5^3 \cdot 5^{-7}\)[/tex]:
[tex]\[
5^3 \cdot 5^{-7} = 5^{3 + (-7)} = 5^{-4}
\][/tex]
So, we are looking for expressions that equal [tex]\(5^{-4}\)[/tex].
### Comparing with the given options:
1. [tex]\(\frac{1}{5^{-4}}\)[/tex]:
[tex]\[
\frac{1}{5^{-4}} = 5^4
\][/tex]
This expression evaluates to [tex]\(5^4\)[/tex], not [tex]\(5^{-4}\)[/tex]. It is not equal to [tex]\(5^{-4}\)[/tex].
2. [tex]\(\frac{1}{5^4}\)[/tex]:
[tex]\[
\frac{1}{5^4}
\][/tex]
This is the reciprocal of [tex]\(5^4\)[/tex], which means:
[tex]\[
\frac{1}{5^4} = 5^{-4}
\][/tex]
This expression is indeed equal to [tex]\(5^{-4}\)[/tex].
3. [tex]\(5^{-4}\)[/tex]:
[tex]\[
5^{-4}
\][/tex]
This is immediately equal to the simplified result, [tex]\(5^{-4}\)[/tex].
4. [tex]\(5^4\)[/tex]:
[tex]\[
5^4
\][/tex]
This is not equal to [tex]\(5^{-4}\)[/tex]. It is the reciprocal of [tex]\(5^{-4}\)[/tex].
5. [tex]\(\frac{1}{-625}\)[/tex]:
[tex]\[
\frac{1}{-625}
\][/tex]
Now, since [tex]\(5^4 = 625\)[/tex], then:
[tex]\[
5^{-4} = \frac{1}{5^4} = \frac{1}{625}
\][/tex]
However, [tex]\(\frac{1}{-625}\)[/tex] is [tex]\(\frac{1}{625}\)[/tex] but negative. Therefore, it is not the same as [tex]\(5^{-4}\)[/tex].
Based on this analysis:
- [tex]\(\frac{1}{5^{-4}}\)[/tex] is .[tex]\(\)[/tex]
- [tex]\(\frac{1}{5^4}\)[/tex] is equal to [tex]\(5^{-4}\)[/tex]
- [tex]\(5^{-4}\)[/tex] is equal to [tex]\(5^{-4}\)[/tex]
- [tex]\(5^4\)[/tex] is not equal to [tex]\(5^{-4}\)[/tex]
- [tex]\(\frac{1}{-625}\)[/tex] is not equal to [tex]\(5^{-4}\)[/tex]
Therefore, the expressions equal to [tex]\(5^3 \cdot 5^{-7}\)[/tex] are:
[tex]\[
\frac{1}{5^4} \quad \text{and} \quad 5^{-4}
\][/tex]
