Find irrational number between a and b

Answer

To find an irrational number between two numbers \( a \) and \( b \), we can use the method of taking the square root of the product of \( a \) and \( b \), provided that the product is not a perfect square. Let's go through the steps using the example of finding an irrational number between \( 5 \) and \( 6 \).

### Step-by-Step Solution:

1. **Identify the two numbers**: In this case, we have \( a = 5 \) and \( b = 6 \).

2. **Calculate the product of \( a \) and \( b \)**:
\[ a \times b = 5 \times 6 = 30 \]

3. **Check if the product is a perfect square**:
- The perfect squares near \( 30 \) are \( 25 \) (which is \( 5^2 \)) and \( 36 \) (which is \( 6^2 \)).
- Since \( 30 \) is not a perfect square, we can proceed.

4. **Take the square root of the product**:
\[ \sqrt{a \times b} = \sqrt{30} \]

5. **Determine the range of the square root**:
- We know that \( 5 < \sqrt{30} < 6 \) because \( 5^2 = 25 < 30 < 36 = 6^2 \).

6. **Conclusion**:
- Thus, \( \sqrt{30} \) is an irrational number that lies between \( 5 \) and \( 6 \).

### Final Answer:
The irrational number between \( 5 \) and \( 6 \) is \( \sqrt{30} \).

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