The problem asks to select all expressions that are equivalent to $4^2 \cdot 4^7$. The options given are $4^{-5}$, $\frac{1}{4^{-5}}$, $\frac{4^3}{4^8}$, and $4^{-14}$.

1. Problem Description

The problem asks to select all expressions that are equivalent to $4^2 \cdot 4^7$. The options given are $4^{-5}$, $\frac{1}{4^{-5}}$, $\frac{4^3}{4^8}$, and $4^{-14}$.

2. Solution Steps

First, we simplify the expression $4^2 \cdot 4^7$ using the rule $a^m \cdot a^n = a^{m+n}$:

$4^2 \cdot 4^7 = 4^{2+7} = 4^9$

Now, we check each option:

* $4^{-5}$: This is not equal to $4^9$.

* $\frac{1}{4^{-5}}$: Using the rule $\frac{1}{a^{-n}} = a^n$, we have $\frac{1}{4^{-5}} = 4^5$. This is not equal to $4^9$.

* $\frac{4^3}{4^8}$: Using the rule $\frac{a^m}{a^n} = a^{m-n}$, we have $\frac{4^3}{4^8} = 4^{3-8} = 4^{-5}$. This is not equal to $4^9$.

* $4^{-14}$: This is not equal to $4^9$.

Based on the provided options, it seems there is an error in transcription. Let's assume there are two other options: $4^9$ and $\frac{1}{4^{-9}}$

Now, we check each option:

* $4^{-5}$: This is not equal to $4^9$.

* $\frac{1}{4^{-5}}$: Using the rule $\frac{1}{a^{-n}} = a^n$, we have $\frac{1}{4^{-5}} = 4^5$. This is not equal to $4^9$.

* $\frac{4^3}{4^8}$: Using the rule $\frac{a^m}{a^n} = a^{m-n}$, we have $\frac{4^3}{4^8} = 4^{3-8} = 4^{-5}$. This is not equal to $4^9$.

* $4^{-14}$: This is not equal to $4^9$.

* $4^9$: This is equal to $4^9$.

* $\frac{1}{4^{-9}}$: Using the rule $\frac{1}{a^{-n}} = a^n$, we have $\frac{1}{4^{-9}} = 4^9$. This is equal to $4^9$.

Given the provided options $4^{-5}$, $\frac{1}{4^{-5}}$, $\frac{4^3}{4^8}$ and $4^{-14}$ none are correct. But we can notice that $\frac{1}{4^{-5}} = 4^5$.

3. Final Answer

None of the given options are equivalent to $4^2 \cdot 4^7 = 4^9$.

However, if we assume that the question asks which expression can be rewritten as the power of 4, then we can transform the expressions as follows:

$4^{-5} = 4^{-5}$

$\frac{1}{4^{-5}} = 4^{5}$

$\frac{4^{3}}{4^{8}} = 4^{-5}$

$4^{-14} = 4^{-14}$

Thus, they are all in the form of $4^x$.

Without more context or confirmation of the options, I am unable to provide a definitive answer. Based on the options given in the question, there are no correct answers. If there was a typo in the expression given, for example, if the target expression was $\frac{4^3}{4^{-6}}$, which would be $4^9$.

Final Answer: No correct options based on the picture provided.