The given expression is:

$4^5 \times 4^{-7} \div 4^{-2}$

We will solve it using the properties of exponents.

1. Multiplication of powers with the same base: When multiplying terms with the same base, you add the exponents:

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

2. Division of powers with the same base: When dividing terms with the same base, you subtract the exponents:

$4^{-2} \div 4^{-2} = 4^{-2 - (-2)} = 4^0$

Since $4^0 = 1$, the correct answer is $4^0$.

The correct option is $4^0$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. What is the rule for multiplying exponents with the same base?
2. How do you simplify division of exponents with the same base?
3. What is the value of any number raised to the power of 0?
4. Can you simplify $4^3 \times 4^{-5}$?
5. How do negative exponents work in mathematical expressions?

Tip: Remember that when exponents are added or subtracted during multiplication or division, you can quickly simplify expressions with the same base!