The question asks to select all expressions that are equivalent to $5^4 \cdot 7^4$.

Let's break down the provided options one by one:

Step 1: Evaluate the original expression

The given expression is: $5^4 \cdot 7^4$ This can be rewritten as: $(5 \cdot 7)^4 = 35^4$ So, the equivalent expression to $5^4 \cdot 7^4$ is $35^4$.

Step 2: Analyze the given options

1. Option 1: $\frac{35^{-3}}{35^{-7}}$
   Applying the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we get: $35^{-3 - (-7)} = 35^{-3 + 7} = 35^4$ This is equivalent to $35^4$.

2. Option 2: $35^{16}$
   This is not equivalent to $35^4$, so it is incorrect.

3. Option 3: $\frac{1}{35^4}$
   This is not equivalent to $35^4$; it is the reciprocal, so it is incorrect.

4. Option 4: $(35^{-4})^{-1}$
   Applying the property $(a^m)^{-1} = a^{-m}$, we get: $(35^{-4})^{-1} = 35^4$ This is equivalent to $35^4$.

Correct options:

• $\frac{35^{-3}}{35^{-7}}$
• $(35^{-4})^{-1}$

Summary:

The expressions equivalent to $5^4 \cdot 7^4$ are:

1. $\frac{35^{-3}}{35^{-7}}$
2. $(35^{-4})^{-1}$

Would you like further details or explanations?

Here are 5 related questions:

1. What are the properties of exponents used in simplifying expressions?
2. How do you simplify expressions involving negative exponents?
3. How can reciprocal properties be applied to exponent rules?
4. What is the significance of combining powers with the same base?
5. How can you convert between expressions like $a^n \cdot b^n$ and $(a \cdot b)^n$?

Tip: Always check if you can rewrite products of powers as powers of products to simplify expressions more easily.