The conjugate product of terms involving square roots results in a rational number because of the algebraic property of conjugates. Specifically, when you have two terms, \( a + \sqrt{b} \) and \( a - \sqrt{b} \), their product can be expressed as:
\[ (a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 \]
Using the difference of squares formula, the right side simplifies to:
\[ a^2 - b \]
Since both \( a^2 \) (the square of a rational number) and \( b \) (a non-negative rational number) are rational numbers, their difference \( a^2 - b \) is also a rational number.
This reasoning can be applied generally to any pair of conjugate terms involving square roots. Therefore, the product of any two conjugate terms that include square roots will always result in a rational number due to the elimination of the square root during multiplication, leading to a form that consists solely of rational numbers.