Question

Find the sum of 3, square root of, 2, end square root3
2
​
and 3, square root of, 25, end square root3
25
​
in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

Answer 1

To find the sum of \(3\sqrt{2}\) and \(3\sqrt{25}\), we first evaluate the individual square roots.

1. Evaluate \(\sqrt{25}\):
\[
\sqrt{25} = 5
\]

2. Now, substitute this value back into the expression:
\[
3\sqrt{25} = 3 \times 5 = 15
\]

3. Now we can add the two terms:
\[
3\sqrt{2} + 3\sqrt{25} = 3\sqrt{2} + 15
\]

4. The expression can be written as:
\[
3\sqrt{2} + 15
\]

This is the sum in its simplest form. To analyze whether the result is rational or irrational, let's look at the component parts:

- The term \(3\sqrt{2}\) is irrational because \(\sqrt{2}\) is an irrational number and multiplying an irrational number by a rational number (3) still yields an irrational number.
- The term \(15\) is rational.

The sum of a rational number (15) and an irrational number (\(3\sqrt{2}\)) is always irrational. Therefore, the result \(3\sqrt{2} + 15\) is irrational.

### Final Answer:
The sum is \(3\sqrt{2} + 15\), and it is irrational.