To find the sum of \( 3\sqrt{3} \) and \( 2\sqrt{3} \), we can start by adding the two terms.
Both terms have a common factor of \( \sqrt{3} \):
\[ 3\sqrt{3} + 2\sqrt{3} = (3 + 2)\sqrt{3} = 5\sqrt{3} \]
Next, we simplify the result, which is \( 5\sqrt{3} \).
Now, let's determine whether \( 5\sqrt{3} \) is rational or irrational.
- A rational number can be expressed as the quotient of two integers, \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).
- An irrational number cannot be expressed in this way.
The square root of 3, \( \sqrt{3} \), is known to be an irrational number. Multiplying an irrational number by a non-zero rational number (like 5) yields another irrational number. Therefore, \( 5\sqrt{3} \) is irrational.
In summary, the sum \( 5\sqrt{3} \) is in its simplest form, and it is an irrational number.