Question

Find the sum of 5, square root of, 5, end square root5
5
​
and square root of, 25, end square root
25
​
in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
Answer

Result:


The result is
because it
be written as the ratio of two integers and its decimal expansion
terminate or repeat.

Answer 1

To find the sum of \( \sqrt{5} \) and \( \sqrt{25} \), we start by calculating each expression:

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

2. **Now, we can sum the two values**:
\[
\sqrt{5} + \sqrt{25} = \sqrt{5} + 5
\]

So, the result of the sum is \( \sqrt{5} + 5 \).

Next, we need to determine whether this result is rational or irrational.

- **Rational numbers** can be expressed as the ratio of two integers, and their decimal expansions either terminate or repeat.
- **Irrational numbers** cannot be expressed as a ratio of integers, and their decimal expansions do not terminate or repeat.

Now we analyze each component:
- \( \sqrt{5} \) is known to be an irrational number.
- \( 5 \) is a rational number (since it can be expressed as \( \frac{5}{1} \)).

When you add a rational number and an irrational number, the result is always irrational. Therefore, \( \sqrt{5} + 5 \) is irrational.

### Final Answer
Result:
The result is \( \sqrt{5} + 5 \) because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.