Rationalizing Denominators: Step-by-Step Guide
Rationalizing a denominator means rewriting a fraction so there is no square root (or other root) in the bottom. You do this by multiplying the top and bottom by the right thing. If the bottom is just a root, multiply by that same root. If the bottom has a root added to or subtracted from a number, multiply by its conjugate (the same two terms with the sign switched).
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Look at the bottom. Is the bottom a single root, or is it a sum or difference with a root in it? The method depends on which one it is.\[\frac{a}{\sqrt{b}} \quad\text{vs.}\quad \frac{a}{c + \sqrt{d}}\]
- For a single root, multiply by that root over itself. Multiply the top and the bottom by the same root. For example, multiply by root b over root b, so the bottom becomes b.\[\frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}\]
- For a sum or difference, multiply by the conjugate. The conjugate of c + root d is c - root d. When you multiply them, the root disappears, because (a + b)(a - b) = a^2 - b^2.\[\frac{a}{c+\sqrt{d}} \cdot \frac{c-\sqrt{d}}{c-\sqrt{d}} = \frac{a(c-\sqrt{d})}{c^{2}-d}\]
- Simplify. Multiply out, combine like terms, and cancel any common factors.\[\text{simplified form, no radical in denominator}\]
Worked example
Rationalize: 5 / (3 - sqrt(2))
Step 1: The bottom, 3 - root 2, is a difference with a root in it, so we use the conjugate.
Step 2: The conjugate of 3 - root 2 is 3 + root 2.
Step 3: Multiply the top and the bottom by 3 + root 2: (5(3 + root 2)) / ((3 - root 2)(3 + root 2)).
Step 4: The bottom: 3^2 - (root 2)^2 = 9 - 2 = 7.
Step 5: The top: 5(3 + root 2) = 15 + 5 root 2.
Step 6: The answer is (15 + 5 root 2) / 7.
Check: 5 / (3 - 1.414) = 5 / 1.586 = about 3.153. And (15 + 5 x 1.414) / 7 = 22.07 / 7 = about 3.153. They match.
Tips and common mistakes
- For a single square root, multiply the top and bottom by that root.
- For other roots, multiply by what you need to make a perfect power. For 1 over the cube root of 2, multiply by the cube root of 4 over the cube root of 4, so the bottom becomes the cube root of 8, which is 2.
- The conjugate of a + b is a - b. Their product is a^2 - b^2, so the roots disappear.
- Multiply the whole top and the whole bottom, never just one part.
- Simplify at the end. You can often cancel common factors.
- Most teachers expect no roots in the denominator for a final answer.
- Be careful with signs when you multiply by the conjugate.