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The question asks, **"Which among the following is an irrational number?"** with four options provided. Let's go through them:

1. **\( \frac{\sqrt{27}}{3} \)**:
   - \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \), so the expression becomes \( \frac{3\sqrt{3}}{3} = \sqrt{3} \).
   - Since \( \sqrt{3} \) is irrational, this option is irrational.

2. **\( (2 + \sqrt{2})^2 + (2 - \sqrt{2})^2 \)**:
   - Expanding these squares, \( (2 + \sqrt{2})^2 = 4 + 4\sqrt{2} + 2 = 6 + 4\sqrt{2} \).
   - Similarly, \( (2 - \sqrt{2})^2 = 4 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2} \).
   - Adding them gives \( (6 + 4\sqrt{2}) + (6 - 4\sqrt{2}) = 12 \), which is a rational number.

3. **\( (2 + \sqrt{2})^2 \)**:
   - Expanding this gives \( 6 + 4\sqrt{2} \), which is irrational because of the \( 4\sqrt{2} \) term.

4. **\( \frac{2.7}{0.3} \)**:
   - This simplifies to 9, which is a rational number.

### Conclusion:
- The irrational numbers among the options are \( \frac{\sqrt{27}}{3} \) and \( (2 + \sqrt{2})^2 \).
- However, since you selected the last option (2.7/0.3), that is **not irrational**, it's rational. The correct irrational number is **\( \frac{\sqrt{27}}{3} \)**.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How can you determine if a number involving square roots is irrational?
2. Why does simplifying square roots often lead to irrational numbers?
3. What makes a number irrational versus rational?
4. How do expansions of binomials like \( (2 + \sqrt{2})^2 \) affect the result's rationality?
5. Can a fraction of two decimal numbers ever result in an irrational number?

**Tip**: Any square root of a non-perfect square is irrational.

Which among the following is an irrational number? (Options: √27/3, (2 + √2)^2 + (2 - √2)^2, (2 + √2)^2, 2.7/0.3)

This problem helps in identifying irrational numbers from given algebraic expressions and fractions. It covers key concepts like square roots and binomial expansions, suitable for students in Grades 9-10.

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