Question

find the sum of 3 radical 9 and 3 radical 16 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

Result: ?

the result is (irrational or rational) because it (can or cannot) be written as the ratio of two integers and its decimal expansion (does or does not) terminate or repeat.

Answer 1

To find the sum of \( 3\sqrt{9} \) and \( 3\sqrt{16} \), we start by simplifying each term:

1. Calculate \( 3\sqrt{9} \):
\[
\sqrt{9} = 3 \quad \Rightarrow \quad 3\sqrt{9} = 3 \cdot 3 = 9
\]

2. Calculate \( 3\sqrt{16} \):
\[
\sqrt{16} = 4 \quad \Rightarrow \quad 3\sqrt{16} = 3 \cdot 4 = 12
\]

Now, we can add them together:
\[
9 + 12 = 21
\]

Thus, the result is:
\[
\text{Result: } 21
\]

Next, we determine whether this result is rational or irrational. The result \( 21 \) is a whole number and can be expressed as a fraction:
\[
21 = \frac{21}{1}
\]

Therefore, it is rational because it can be written as the ratio of two integers (21 and 1), and its decimal expansion does terminate (21.0).

So, we can summarize the findings as follows:

The result is **rational** because it **can** be written as the ratio of two integers, and its decimal expansion **does** terminate.