Rationalization

Today, we’re going to learn about a process called rationalization. Can anyone tell me why we might want to eliminate square roots from the denominator of a fraction?

Maybe it makes the calculations easier?

Exactly! Having a square root in the denominator can complicate calculations. So, what do we do instead?

Do we multiply by something?

Yes, we multiply by the conjugate. Let’s look at an example: how would we rationalize $\frac{1}{\sqrt{2}}$?

We can multiply both the numerator and denominator by $\sqrt{2}$!

Great! So, $\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}$ becomes $\frac{\sqrt{2}}{2}$. We’ve eliminated the square root from the denominator!

Now, what if our denominator was a binomial surd like $\sqrt{2} + 1$? What should we multiply by here?

We can multiply by its conjugate, so $\sqrt{2} - 1$!

Correct! So how do we rationalize $\frac{1}{\sqrt{2} + 1}$?

We multiply the numerator and denominator by $\sqrt{2} - 1$.

Exactly! This gives us $\frac{\sqrt{2} - 1}{\sqrt{2}^2 - 1^2}$, simplifying to $\frac{\sqrt{2} - 1}{1} = \sqrt{2} - 1$.