Rationalizing the Denominator

Why rationalize?

Convention says the denominator should have no radicals. Rationalizing clears them out.

Step-by-step: Simplify $\dfrac{5}{\sqrt{3} - 1}$

Step 1 — Identify the conjugate. The conjugate of $(\sqrt{3} - 1)$ is $(\sqrt{3} + 1)$ — same terms, opposite sign.

Step 2 — Multiply top and bottom by the conjugate (this equals multiplying by 1):

$\frac{5}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1}$

Step 3 — Multiply the denominator using $(a-b)(a+b) = a^2 - b^2$:

$(\sqrt{3})^2 - (1)^2 = 3 - 1 = 2$

The radical is gone from the denominator!

Step 4 — Multiply the numerator:

$5(\sqrt{3} + 1) = 5\sqrt{3} + 5$

Step 5 — Final answer:

$\frac{5\sqrt{3} + 5}{2}$

Check: $\frac{5}{\sqrt{3}-1} \approx \frac{5}{0.732} \approx 6.830$. $\frac{5\sqrt{3}+5}{2} = \frac{8.660+5}{2} = 6.830$ ✓

Simple case (single radical)

For $\frac{5}{\sqrt{3}}$: multiply by $\frac{\sqrt{3}}{\sqrt{3}}$: $\frac{5\sqrt{3}}{3}$.

Try it in the visualization

Adjust the numerator, radical, and constant. The conjugate multiplication is animated step by step. A decimal check verifies the answer.