To find the product of \(2\sqrt{15}\) and \(3\sqrt{15}\), we can start by multiplying these two expressions.
\[ 2\sqrt{15} \times 3\sqrt{15} \]
We can rearrange the multiplication as follows:
\[ (2 \times 3) \times (\sqrt{15} \times \sqrt{15}) \]
Calculating each part, we first find:
\[ 2 \times 3 = 6 \]
For the square roots, we use the property that \(\sqrt{a} \times \sqrt{a} = a\):
\[ \sqrt{15} \times \sqrt{15} = 15 \]
Now we multiply these results together:
\[ 6 \times 15 = 90 \]
Thus, the product of \(2\sqrt{15}\) and \(3\sqrt{15}\) is:
\[ 90 \]
Next, we determine whether the result is rational or irrational. The number \(90\) is an integer, and all integers are rational numbers since they can be expressed as the quotient of two integers (for example, \(90\) can be expressed as \(\frac{90}{1}\)).
Therefore, the final result is:
- The product is \(90\).
- The result is a rational number.