To determine which of the given expressions are equivalent to [tex]\(\frac{4^{-3}}{4^{-8}}\)[/tex], let's start by simplifying the original expression using the properties of exponents.
The given expression is [tex]\(\frac{4^{-3}}{4^{-8}}\)[/tex]. According to the properties of exponents:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
Applying this property, we have:
[tex]\[
\frac{4^{-3}}{4^{-8}} = 4^{-3 - (-8)} = 4^{-3 + 8} = 4^5
\][/tex]
So the expression [tex]\(\frac{4^{-3}}{4^{-8}}\)[/tex] simplifies to [tex]\(4^5\)[/tex].
Now, let's evaluate which of the given expressions are equivalent to [tex]\(4^5\)[/tex]:
1. [tex]\(\frac{4^8}{4^3}\)[/tex]:
[tex]\[
\frac{4^8}{4^3} = 4^{8-3} = 4^5
\][/tex]
This is equivalent.
2. [tex]\(\frac{4^{-8}}{4^{-3}}\)[/tex]:
[tex]\[
\frac{4^{-8}}{4^{-3}} = 4^{-8 - (-3)} = 4^{-8 + 3} = 4^{-5}
\][/tex]
This is not equivalent to [tex]\(4^5\)[/tex].
3. [tex]\(4^{-5}\)[/tex]:
This is clearly not equivalent to [tex]\(4^5\)[/tex].
4. [tex]\(4^5\)[/tex]:
This is exactly the same as our simplified result.
Therefore, the expressions that are equivalent to [tex]\(\frac{4^{-3}}{4^{-8}}\)[/tex] are:
[tex]\[
\boxed{\frac{4^8}{4^3} \text{ and } 4^5}
\][/tex]
The corresponding indices of these expressions are:
[tex]\[
\boxed{1 \text{ and } 4}
\][/tex]
