Let's determine whether the numbers 39 and 104 are perfect squares by following the steps mentioned earlier.

For $39$:

1. Calculate the Square Root:

   • $\sqrt{39} \approx 6.244$

2. Check if it's an Integer:

   • $6.244$ is not an integer.

3. Check by Squaring:

   • The closest integers to $\sqrt{39}$ are 6 and 7.
   • $6^2 = 36$
   • $7^2 = 49$

Since $36 < 39 < 49$, $39$ is not a perfect square.

For $104$:

1. Calculate the Square Root:

   • $\sqrt{104} \approx 10.198$

2. Check if it's an Integer:

   • $10.198$ is not an integer.

3. Check by Squaring:

   • The closest integers to $\sqrt{104}$ are 10 and 11.
   • $10^2 = 100$
   • $11^2 = 121$

Since $100 < 104 < 121$, $104$ is not a perfect square.

In conclusion, neither 39 nor 104 are perfect squares as their square roots are not integers, and the closest integer squares do not match the original numbers.

Do you have any questions or need further details on any part of the process?

Follow-Up Questions:

1. How can you quickly check if a large number is a perfect square using a calculator?
2. Are there any patterns in the distribution of perfect squares among natural numbers?
3. How do perfect squares relate to geometric shapes, such as squares?
4. What are some applications of perfect squares in real life?
5. Can a fraction be a perfect square, and how would you determine that?
6. How does understanding perfect squares help in solving quadratic equations?
7. Are there any historical methods for determining if a number is a perfect square?
8. How can the concept of perfect squares be extended to three dimensions?

Tip:

When determining if a number is a perfect square, remember that perfect squares only appear at the square of integers. Checking the approximate square root and comparing with the closest integer squares is a reliable method.