Conjugate Pairs

Multiply $\left(4-\sqrt{3}\right)\left(4+\sqrt{3}\right)$. The four products are 16, $4\sqrt{3}$, $-4\sqrt{3}$ and -3. The two middle terms are opposites and cancel, so what is left is 16 minus 3, which is 13. No root survives.

This always happens. The brackets $a-\sqrt{b}$ and $a+\sqrt{b}$ differ only in the sign between their terms; they are called conjugates. Their product is a difference of squares:

$$\left(a-\sqrt{b}\right)\left(a+\sqrt{b}\right) = {a}^{2}-b$$

That is the reason conjugates matter: when a and b are whole numbers, they turn an expression with a root into a whole number. In the next chapter this is how a root is cleared from the bottom of a fraction.