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Let's analyze and solve the given problems in the image.

### Problem 1:
Determine if the following is a "non-perfect square" number.
- a) 16
- b) 48
- c) 33
- d) 2025

#### Solution:
A perfect square is a number that is the square of an integer.

- \(16 = 4^2\), so 16 is a perfect square.
- \(48\) is not a perfect square because there is no integer \(n\) such that \(n^2 = 48\).
- \(33\) is not a perfect square because there is no integer \(n\) such that \(n^2 = 33\).
- \(2025 = 45^2\), so 2025 is a perfect square.

So, the non-perfect square numbers are \(48\) and \(33\).

### Problem 2:
Determine whether each number is irrational or rational.
- a) \(2\sqrt{7}\)
- b) \(1.121221222\)
- c) \(-\sqrt{81}\)

#### Solution:
- \(2\sqrt{7}\): \(\sqrt{7}\) is an irrational number, and multiplying an irrational number by a rational number (2) results in an irrational number. Hence, \(2\sqrt{7}\) is irrational.
- \(1.121221222\): This number does not repeat or terminate, indicating it is a non-terminating, non-repeating decimal, making it irrational.
- \(-\sqrt{81}\): \(\sqrt{81} = 9\), and \(-9\) is a rational number.

### Summary:
1. **Non-perfect square numbers**: \(b) 48\) and \(c) 33\).
2. **Irrational/Rational**:
   - a) \(2\sqrt{7}\) is irrational.
   - b) \(1.121221222\) is irrational.
   - c) \(-\sqrt{81}\) is rational.

Do you want more details on any of these problems or have any questions?

Here are five related questions to expand your understanding:
1. What makes a number a perfect square?
2. Can the product of two irrational numbers be rational? Provide an example.
3. How can you determine if a large number is a perfect square?
4. Why are non-terminating, non-repeating decimals considered irrational?
5. Is the square root of any prime number irrational? Why or why not?

**Tip:** When determining if a number is a perfect square, check if its square root is an integer. If the square root is not an integer, the number is not a perfect square.

1. Determine if the following is a 'non-perfect square' number: a) 16, b) 48, c) 33, d) 2025. 2. Determine whether each number is irrational or rational: a) 2√7, b) 1.121221222, c) -√81.

Solve problems on identifying non-perfect squares and determining whether numbers are rational or irrational. This problem includes the identification of non-perfect squares such as 48 and 33, and analyzing numbers like 2√7, 1.121221222, and -√81 to classify them as rational or irrational. Suitable for Grades 6-8.

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