The problem in the image asks which expressions are equivalent to $7 \cdot 7 \cdot 7 \cdot 7 \cdot 7$ (which is $7^5$).

Let's evaluate each option to see which is equivalent to $7^5$:

1. Option a: $\frac{7^8}{7^2} = 7^{8-2} = 7^6$ This is not equal to $7^5$.

2. Option b: $7^6 \cdot 7^1 = 7^{6+1} = 7^7$ This is also not equal to $7^5$.

3. Option c: $(7^2)^3 = 7^{2 \cdot 3} = 7^6$ This is not equal to $7^5$.

4. Option d: $\frac{7^{12}}{7^2} = 7^{12-2} = 7^{10}$ This is not equal to $7^5$.

None of the provided answers seem to directly match $7^5$, though the checkmark indicates that option a is selected, which simplifies to $7^6$, but is one exponent off from $7^5$.

Let me know if you need more clarification, or if you'd like me to explore further! Here are some related questions:

1. How do we simplify expressions involving exponents in division?
2. What are the properties of exponents in multiplication and division?
3. Can you explain how to combine powers of the same base?
4. What is the general formula for simplifying $a^m \cdot a^n$?
5. What is the difference between multiplying powers and raising a power to another power?

Tip: When simplifying exponents, always check if you are dealing with multiplication, division, or powers raised to powers, as each has different rules!