Let's evaluate each of the options to determine which one is an irrational number:

A. 0.333...
This number is a repeating decimal, which means it can be expressed as a fraction, specifically $\frac{1}{3}$. Since it can be written as a fraction, it is a rational number.

B. $\sqrt{8}$
This is the square root of 8. We know that $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$, and since $\sqrt{2}$ is an irrational number, $2\sqrt{2}$ is also irrational. Therefore, $\sqrt{8}$ is an irrational number.

C. $\sqrt{49}$
The square root of 49 is exactly 7, which is a whole number. Since whole numbers are rational, $\sqrt{49} = 7$ is a rational number.

D. $\frac{5}{22}$
This is a fraction, and any fraction where both the numerator and denominator are integers is a rational number. Therefore, $\frac{5}{22}$ is a rational number.

Conclusion:

The irrational number is B. $\sqrt{8}$.

Would you like more explanation or clarification on this topic?

Additional Questions:

1. What is the definition of a rational number?
2. How can you determine if a square root is rational or irrational?
3. Why is $\sqrt{2}$ considered irrational?
4. Can irrational numbers be expressed as fractions in any form?
5. What is the difference between rational and irrational numbers in terms of their decimal expansions?

Tip: If you ever have trouble deciding if a number is rational or irrational, remember that rational numbers can always be written as fractions, while irrational numbers cannot.