Question

Find the sum of 5, square root of, 25, end square root5
25
​
and square root of, 7, end square root
7
​
in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
Answer
Attempt 1 out of 5

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 \( \sqrt{25} \) and \( \sqrt{7} \), we first need to calculate \( \sqrt{25} \):

\[
\sqrt{25} = 5
\]

Now we can add \( 5 \) and \( \sqrt{7} \):

\[
5 + \sqrt{7}
\]

The sum \( 5 + \sqrt{7} \) cannot be simplified further, so the simplest form of the result is \( 5 + \sqrt{7} \).

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

- The number \( 5 \) is a rational number since it can be expressed as the ratio of two integers (e.g., \( \frac{5}{1} \)).
- The number \( \sqrt{7} \) is an irrational number because it cannot be expressed as a fraction of two integers (its decimal representation is non-terminating and non-repeating).

When you add a rational number to an irrational number, the result is always irrational. Therefore, \( 5 + \sqrt{7} \) is irrational.

In conclusion:

**The result is \( 5 + \sqrt{7} \) because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.**