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Let's analyze each number to determine if it is rational or irrational, and then order them from least to greatest.

### Step 1: Determine if the numbers are rational or irrational

1. **2.777... (repeating decimal)**:
   - This is a repeating decimal (the "..." indicates that the 7 repeats indefinitely).
   - Repeating decimals are always **rational numbers** because they can be expressed as fractions.
   
2. **√8 (square root of 8)**:
   - The square root of 8 is approximately 2.828427..., which is a non-terminating, non-repeating decimal.
   - Non-terminating, non-repeating decimals are **irrational numbers**.

3. **17/9 (fraction)**:
   - This is a simple fraction, and fractions of integers are always **rational numbers**.

### Step 2: Order from least to greatest

We need to compare the approximate values of these numbers:

- **2.777...** (as a repeating decimal) can be written as \( \frac{25}{9} \), which is approximately **2.777**.
- **√8** is approximately **2.828**.
- **17/9** is approximately **1.888**.

### Ordering:
From least to greatest:
- \( \frac{17}{9} \) ≈ 1.888
- 2.777...
- \( \sqrt{8} \) ≈ 2.828

### Final Answer:
- Rational numbers: 2.777... and 17/9
- Irrational number: √8
- Ordered from least to greatest: \( \frac{17}{9} \), 2.777..., \( \sqrt{8} \).

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Would you like more details on any part of the explanation?

Here are 5 related questions for practice:
1. What is the difference between rational and irrational numbers?
2. How can you convert repeating decimals into fractions?
3. What are the properties of irrational numbers?
4. Can every square root be an irrational number?
5. How do you approximate square roots of non-perfect squares?

**Tip**: Any decimal that repeats or terminates can always be expressed as a rational number.

Determine if each number is rational or irrational: 2.777..., √8, and 17/9. Order the three numbers from least to greatest.

This problem involves identifying rational and irrational numbers from 2.777..., √8, and 17/9. Learn how to categorize these numbers and arrange them in order from least to greatest using approximations and definitions. Suitable for Grades 6-8.

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