To determine which of the following expressions is equivalent to [tex]\(5^{-4}\)[/tex], we'll analyze each option step by step. We already know the answers through numerical values that I have verified.
1. Option 1: [tex]\(5^{-4}\)[/tex]
[tex]\[5^{-4}\][/tex]
This directly gives us [tex]\(0.0016\)[/tex], since we know that [tex]\(5^{-4}\)[/tex] simplifies to [tex]\(\frac{1}{5^4} = \frac{1}{625} = 0.0016\)[/tex].
2. Option 2: [tex]\(5^4\)[/tex]
[tex]\[5^4\][/tex]
This is [tex]\(625\)[/tex]. Clearly, [tex]\(625\)[/tex] is not equivalent to [tex]\(0.0016\)[/tex].
3. Option 3: [tex]\(\sqrt[7]{5^3}\)[/tex]
[tex]\[\sqrt[7]{5^3} = (5^3)^{1/7} = 5^{3/7}\][/tex]
This gives us approximately [tex]\(1.993\)[/tex]. This value is not equivalent to [tex]\(0.0016\)[/tex].
4. Option 4: [tex]\(\sqrt[3]{5^7}\)[/tex]
[tex]\[\sqrt[3]{5^7} = (5^7)^{1/3} = 5^{7/3}\][/tex]
This gives us approximately [tex]\(42.749\)[/tex]. This value is not equivalent to [tex]\(0.0016\)[/tex].
By examining each option, we can see that the only expression that matches [tex]\(0.0016\)[/tex] is [tex]\(5^{-4}\)[/tex]. Thus, the correct choice is:
[tex]\[ \boxed{5^{-4}} \][/tex]
