We are tasked with finding between which two integer values the expression $3\sqrt{7}$ lies. To solve this, we first approximate the value of $\sqrt{7}$.

The square root of 7 is approximately: $\sqrt{7} \approx 2.64575$

Now, multiply this by 3: $3\sqrt{7} \approx 3 \times 2.64575 = 7.93725$

Thus, $3\sqrt{7}$ is approximately 7.93725, which means the value lies between 7 and 8.

Conclusion:

The expression $3\sqrt{7}$ is between the values 7 and 8.

Would you like more details on how square roots are approximated, or do you have further questions?

Here are 5 related questions to consider:

1. How do you calculate square roots manually?
2. What is the approximation of $\sqrt{5}$?
3. What is the value of $4\sqrt{10}$?
4. How can we use inequalities to show bounds of expressions involving square roots?
5. How do irrational numbers like $\sqrt{7}$ differ from rational numbers?

Tip: When approximating square roots, you can always square nearby integers to estimate the closest possible values.