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

We need to simplify the given expression $4^2 \cdot 4^{-7}$ using the property of exponents.

When multiplying exponents with the same base, we add the exponents.

$a^m \cdot a^n = a^{m+n}$

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

Now we will analyze each of the options.

The first option is $4^{-5}$. This is equivalent to our simplified expression.

The second option is $\frac{1}{4^{-5}}$. This is equal to $4^5$. This is not equivalent.

The third option is $\frac{4^3}{4^8}$. Using the property $\frac{a^m}{a^n} = a^{m-n}$, we get

$\frac{4^3}{4^8} = 4^{3-8} = 4^{-5}$. This is equivalent to our simplified expression.

The fourth option is $4^{-14}$. This is not equivalent.

3. Final Answer

The equivalent expressions are $4^{-5}$ and $\frac{4^3}{4^8}$.