What is the smallest number by which `sqrt(5)-sqrt(2)` is to be multiplied to make it a rational number? Also find the number so obtained?

Answer

To solve the problem of finding the smallest number by which \( \sqrt{5} - \sqrt{2} \) must be multiplied to make it a rational number, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Expression**: We start with the expression \( \sqrt{5} - \sqrt{2} \).

2. **Rationalization Concept**: To convert this expression into a rational number, we can use the technique of rationalization. This involves multiplying by the conjugate of the expression. The conjugate of \( \sqrt{5} - \sqrt{2} \) is \( \sqrt{5} + \sqrt{2} \).

3. **Multiply by the Conjugate**: We multiply \( \sqrt{5} - \sqrt{2} \) by its conjugate:
\[ (\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) \]

4. **Apply the Difference of Squares Formula**: The product can be simplified using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):
\[ = (\sqrt{5})^2 - (\sqrt{2})^2 \]

5. **Calculate the Squares**: Now compute the squares:
\[ = 5 - 2 = 3 \]

6. **Conclusion**: Thus, the smallest number by which \( \sqrt{5} - \sqrt{2} \) should be multiplied to obtain a rational number is \( \sqrt{5} + \sqrt{2} \), and the resulting rational number is \( 3 \).

### Final Answer:
The smallest number to multiply \( \sqrt{5} - \sqrt{2} \) is \( \sqrt{5} + \sqrt{2} \), and the rational number obtained is \( 3 \).