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.