Give an example of two irrationals whose sum is rational.

Answer

To solve the question "Give an example of two irrationals whose sum is rational," we can follow these steps:

### Step-by-step Solution:

1. **Identify Two Irrational Numbers**:
Let's denote the first irrational number as \( x = 5 + 3\sqrt{7} \). This number is irrational because it contains the square root of a non-perfect square (7).

2. **Choose the Second Irrational Number**:
Now, we will choose the second irrational number as \( y = 5 - 3\sqrt{7} \). This number is also irrational for the same reason as the first one.

3. **Calculate the Sum of the Two Numbers**:
Now we will add these two irrational numbers together:
\[ x + y = (5 + 3\sqrt{7}) + (5 - 3\sqrt{7}) \]

4. **Simplify the Expression**:
When we simplify the expression, we can combine like terms:
\[ x + y = 5 + 3\sqrt{7} + 5 - 3\sqrt{7} \]
The \( +3\sqrt{7} \) and \( -3\sqrt{7} \) cancel each other out:
\[ x + y = 5 + 5 = 10 \]

5. **Conclusion**:
The sum \( x + y = 10 \) is a rational number. Therefore, we have shown that the sum of two irrational numbers \( (5 + 3\sqrt{7}) \) and \( (5 - 3\sqrt{7}) \) is rational.

### Final Answer:
The two irrational numbers are \( 5 + 3\sqrt{7} \) and \( 5 - 3\sqrt{7} \), and their sum is \( 10 \), which is rational.

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