The question in the image asks to find the tens digit of $7^{707}$.

Since $7^{707}$ is a very large number, finding the tens digit directly through computation would be impractical. Instead, we can use modular arithmetic to find the last two digits of $7^{707}$ (mod 100). Once we find the last two digits, the tens digit will be the second-to-last digit.

Step 1: Break the problem into modulo 100

We want to calculate: $7^{707} \mod 100$

We can use properties of modular exponentiation to simplify the calculation. Applying repeated squaring and modular reductions can give us the last two digits of $7^{707}$.

Let's calculate this.The last two digits of $7^{707}$ are 43. Therefore, the tens digit is 4.

However, the question asks for the tens digit among the provided choices, which are 7, 8, 9, and 6. Based on the problem, it seems that the correct tens digit according to a certain modulo strategy or approach should have aligned with one of these choices, likely due to a unique interpretation or approximation typically taught in some curricula.

Would you like more details on a different possible interpretation of the problem?

Here are five related questions you might find useful:

1. How do we calculate large powers modulo a number?
2. What is the significance of modular arithmetic in number theory?
3. Can the last two digits of other powers of 7 be generalized?
4. What role does number theory play in cryptography?
5. How can repeated squaring help in computing large powers efficiently?

Tip: Modular exponentiation is a powerful tool for simplifying very large number computations, especially when only specific digits (like the last few) are needed.