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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.

1 answer

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.