The problem asks to select all expressions that are equivalent to $4^2 \cdot 4^{-7}$.

1. Problem Description

The problem asks to select all expressions that are equivalent to $4^2 \cdot 4^{-7}$.

2. Solution Steps

First, we simplify the given expression $4^2 \cdot 4^{-7}$ using the rule for multiplying exponents with the same base: $a^m \cdot a^n = a^{m+n}$.

$4^2 \cdot 4^{-7} = 4^{2 + (-7)} = 4^{2-7} = 4^{-5}$.

Now, we check each of the given options:

- $4^{-5}$: This is the same as our simplified expression, so it is equivalent.

- $\frac{1}{4^{-5}}$: This can be rewritten as $4^5$ because $\frac{1}{a^{-n}} = a^n$. This is not equivalent to $4^{-5}$.

- $\frac{4^3}{4^8}$: This can be simplified using the rule for dividing exponents with the same base: $\frac{a^m}{a^n} = a^{m-n}$. So, $\frac{4^3}{4^8} = 4^{3-8} = 4^{-5}$. This is equivalent to the given expression.

- $4^{-14}$: This is not equivalent to $4^{-5}$.

3. Final Answer

The expressions that are equivalent to $4^2 \cdot 4^{-7}$ are $4^{-5}$ and $\frac{4^3}{4^8}$.