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Let's analyze the expressions to determine if they are irrational numbers. A number is **irrational** if it cannot be written as a fraction of two integers, meaning it often involves non-repeating, non-terminating decimals like square roots of non-perfect squares.

### Option 1: \((\sqrt{8} + \sqrt{2})(\sqrt{12} - \sqrt{3})\)
- We can expand this expression using the distributive property (FOIL):
  \[
  (\sqrt{8})(\sqrt{12}) - (\sqrt{8})(\sqrt{3}) + (\sqrt{2})(\sqrt{12}) - (\sqrt{2})(\sqrt{3})
  \]
  Simplifying the square roots:
  \[
  \sqrt{8 \times 12} - \sqrt{8 \times 3} + \sqrt{2 \times 12} - \sqrt{2 \times 3}
  = \sqrt{96} - \sqrt{24} + \sqrt{24} - \sqrt{6}
  \]
  We know that \(\sqrt{96}, \sqrt{24}, \sqrt{6}\) are all irrational numbers, meaning the result remains irrational.

### Option 2: \((\sqrt{8} - \sqrt{2})(\sqrt{18} + \sqrt{2})\)
- Again, expand this using FOIL:
  \[
  (\sqrt{8})(\sqrt{18}) + (\sqrt{8})(\sqrt{2}) - (\sqrt{2})(\sqrt{18}) - (\sqrt{2})(\sqrt{2})
  \]
  Simplifying the square roots:
  \[
  \sqrt{8 \times 18} + \sqrt{8 \times 2} - \sqrt{2 \times 18} - 2
  = \sqrt{144} + \sqrt{16} - \sqrt{36} - 2
  = 12 + 4 - 6 - 2 = 8
  \]
  Since the result is 8, this is a rational number.

### Option 3: \(\frac{\sqrt{6} + \sqrt{3}}{\sqrt{6} - \sqrt{3}}\)
- Rationalizing the denominator:
  \[
  \frac{(\sqrt{6} + \sqrt{3})(\sqrt{6} + \sqrt{3})}{(\sqrt{6} - \sqrt{3})(\sqrt{6} + \sqrt{3})}
  = \frac{(\sqrt{6})^2 + 2\sqrt{6}\sqrt{3} + (\sqrt{3})^2}{(\sqrt{6})^2 - (\sqrt{3})^2}
  \]
  Simplifying:
  \[
  = \frac{6 + 2\sqrt{18} + 3}{6 - 3} = \frac{9 + 6\sqrt{2}}{3} = 3 + 2\sqrt{2}
  \]
  Since \(2\sqrt{2}\) is irrational, the entire expression is irrational.

### Option 4: \(\frac{\sqrt{8} + \sqrt{2}}{\sqrt{8} - \sqrt{2}}\)
- Rationalizing the denominator:
  \[
  \frac{(\sqrt{8} + \sqrt{2})(\sqrt{8} + \sqrt{2})}{(\sqrt{8} - \sqrt{2})(\sqrt{8} + \sqrt{2})}
  = \frac{(\sqrt{8})^2 + 2\sqrt{8}\sqrt{2} + (\sqrt{2})^2}{(\sqrt{8})^2 - (\sqrt{2})^2}
  \]
  Simplifying:
  \[
  = \frac{8 + 2\sqrt{16} + 2}{8 - 2} = \frac{8 + 8 + 2}{6} = \frac{18}{6} = 3
  \]
  Since the result is 3, this is a rational number.

### Conclusion:
- **Irrational numbers**: Option 1 and Option 3.
  
Would you like further details on any of these steps?

### Related questions:
1. What is the significance of irrational numbers in algebra?
2. How do you simplify expressions involving square roots?
3. What is the difference between rationalizing the numerator vs. the denominator?
4. How can you identify if a number is irrational without expanding expressions?
5. How does the distributive property work when dealing with square roots?

### Tip:
When simplifying square root expressions, always check if the roots can simplify to integers, which can help determine if the overall expression is rational or irrational.

Which of the following are irrational numbers?
1) (√8 + √2)(√12 - √3)
2) (√8 - √2)(√18 + √2)
3) (√6 + √3) / (√6 - √3)
4) (√8 + √2) / (√8 - √2)

Learn how to identify irrational numbers in expressions involving square roots. This example includes step-by-step solutions, using concepts like simplification, distributive property, and rationalization. Suitable for students in Grades 9-12.

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