To determine whether a given number is a perfect square, we need to check if there exists an integer $n$ such that $n^2$ equals the given number. Here’s how to approach the problem for different numbers:

Given Numbers:

Let's assume the numbers are:

1. $n_1$
2. $n_2$
3. $n_3$

(Please provide specific numbers if you have them in mind. For now, I'll illustrate the process using placeholders.)

Steps to Determine if a Number is a Perfect Square:

1. Check the Square Root:

   • Calculate the square root of each number.
   • Check if the square root is an integer.

2. Check by Squaring Integers:

   • Find the integer part of the square root.
   • Square this integer to see if it matches the original number.

Example Process:

Let's demonstrate this with an example number: $50$.

1. Calculate the Square Root:

   • $\sqrt{50} \approx 7.071$

2. Check if it's an Integer:

   • $7.071$ is not an integer.

3. Check by Squaring:

   • The closest integers to $\sqrt{50}$ are 7 and 8.
   • $7^2 = 49$
   • $8^2 = 64$

Neither $49$ nor $64$ equals $50$, so $50$ is not a perfect square.

Formal Proof for General Numbers:

Let's generalize this method for any number $n$.

1. Calculate $\sqrt{n}$:

   • Let $k = \lfloor \sqrt{n} \rfloor$ where $\lfloor \cdot \rfloor$ denotes the floor function (integer part).

2. Check the Squared Values:

   • Calculate $k^2$ and $(k+1)^2$.

3. Comparison:

   • If $k^2 \neq n$ and $(k+1)^2 \neq n$, then $n$ is not a perfect square.

Example: Proving Specific Numbers:

Let's consider these specific numbers: $20, 30, 42$.

1. For $20$:

   • $\sqrt{20} \approx 4.472$
   • $4^2 = 16$
   • $5^2 = 25$
   • Since $16 < 20 < 25$, $20$ is not a perfect square.

2. For $30$:

   • $\sqrt{30} \approx 5.477$
   • $5^2 = 25$
   • $6^2 = 36$
   • Since $25 < 30 < 36$, $30$ is not a perfect square.

3. For $42$:

   • $\sqrt{42} \approx 6.480$
   • $6^2 = 36$
   • $7^2 = 49$
   • Since $36 < 42 < 49$, $42$ is not a perfect square.

By following these steps, you can show that any given number is not a perfect square if its square root is not an integer and the integer part squared does not match the number.

Do you have specific numbers you want me to verify, or do you have any questions about this process?

Follow-Up Questions:

1. What are some common properties of perfect squares?
2. How can you determine if a large number is a perfect square without a calculator?
3. What are some examples of perfect squares?
4. How do you simplify the square root of a non-perfect square number?
5. How can you estimate the square root of a non-perfect square number?
6. What is the significance of the floor function in determining perfect squares?
7. Can a negative number be a perfect square?
8. How can you use prime factorization to determine if a number is a perfect square?

Tip:

When checking if a number is a perfect square, remember that the last digit of the square of an integer can only be 0, 1, 4, 5, 6, or 9. This quick check can save time.