To find the product \(3\sqrt{26}\) and \(5\sqrt{26}\), we'll start by multiplying these two expressions together.
The product can be calculated as follows:
\[ (3\sqrt{26})(5\sqrt{26}) = 3 \times 5 \times \sqrt{26} \times \sqrt{26} \]
Calculating the constants and simplifying \( \sqrt{26} \times \sqrt{26} \):
\[ 3 \times 5 = 15 \] \[ \sqrt{26} \times \sqrt{26} = 26 \]
Thus, we have:
\[ (3\sqrt{26})(5\sqrt{26}) = 15 \times 26 \]
Now calculate \(15 \times 26\):
\[ 15 \times 26 = 390 \]
Now we have the product in simplest form:
\[ 3\sqrt{26} \times 5\sqrt{26} = 390 \]
Next, we evaluate whether the result is rational or irrational. The result \(390\) is a whole number, which is a rational number because it can be expressed as a ratio of two integers:
\[ 390 = \frac{390}{1} \]
Therefore, the product of \(3\sqrt{26}\) and \(5\sqrt{26}\) is \(390\), and it is a rational number.