Rationalize Denominators: Remove Radicals from Fractions

Math Made Friendly

Rationalize Denominators: Eliminating Radicals from the Bottom

Learn how to move radicals out of denominators while keeping fractions equivalent.

The Big Idea

Rationalizing a denominator means rewriting a fraction so that no radical remains in the denominator. You do not change the value of the fraction. Instead, you multiply by a carefully chosen form of 1, such as √b ÷ √b or a conjugate divided by itself.

The Main Rule

1 ÷ √b = √b ÷ b, when b > 0

Multiply the numerator and denominator by the same radical. The denominator becomes √b × √b = b, so the radical disappears from the bottom.

Memory tip: Multiply by the missing radical. If the denominator has √5, multiply by √5. If it has a binomial such as 2 + √3, use its conjugate, 2 − √3.

Step-by-Step Method

1. Identify the radical in the denominator.
2. If the denominator is a single square root, multiply the numerator and denominator by that same root.
3. If the denominator is a sum or difference containing a square root, multiply by the conjugate. Change the sign between the two terms.
4. Use the product rule for radicals and simplify.
5. Check that the final denominator contains no radical.

Worked Examples

Example 1: Rationalize 3 ÷ √5

Multiply by √5 ÷ √5. The denominator becomes √5 × √5 = 5.

3 ÷ √5 = (3 × √5) ÷ (√5 × √5) = 3√5 ÷ 5

Example 2: Rationalize 7 ÷ (2√3)

First rewrite the denominator as 2√3. Multiply by √3 ÷ √3.

7 ÷ (2√3) = (7√3) ÷ (2 × 3) = 7√3 ÷ 6

Example 3: Use a conjugate

For 1 ÷ (2 + √3), the conjugate is 2 − √3. Multiply by the conjugate over itself. The denominator follows the difference-of-squares pattern.

1 ÷ (2 + √3) = (2 − √3) ÷ ((2 + √3)(2 − √3)) = (2 − √3) ÷ (4 − 3) = 2 − √3

Example 4: A binomial numerator

Use the conjugate of √5 + 1, which is √5 − 1. Then simplify the denominator.

(√5 + 1) ÷ (√5 − 1) = ((√5 + 1)(√5 + 1)) ÷ ((√5 − 1)(√5 + 1)) = (6 + 2√5) ÷ 4 = (3 + √5) ÷ 2

Example 5: A cube-root denominator

For a denominator containing a cube root, multiply by enough copies of the cube root to create a perfect cube.

1 ÷ cube root of 2 = (cube root of 4) ÷ (cube root of 8) = (cube root of 4) ÷ 2

A Quick Reference Table

On a small phone screen, swipe horizontally inside the table if needed.

Denominator	Multiply by	Result
√7	√7 ÷ √7	Denominator becomes 7
3√2	√2 ÷ √2	Denominator becomes 6
4 + √5	(4 − √5) ÷ (4 − √5)	Denominator becomes 11
cube root of 3	(cube root of 9) ÷ (cube root of 9)	Denominator becomes 3

Common Mistakes to Avoid

• Changing only the denominator: Multiplying the bottom by a radical without multiplying the top changes the value of the fraction.
• Using the same sign with a binomial: For 2 + √3, use the conjugate 2 − √3, not 2 + √3.
• Forgetting to distribute: Expand both factors when multiplying binomials.
• Stopping with a radical below: Simplify until the denominator has no radical.
• Calling any equivalent fraction rationalized: The goal is specifically to remove irrational radicals from the denominator.

Practice Questions

Try these before opening the answer section. On a small phone screen, swipe horizontally inside an equation box if needed.

1. Rationalize 1 ÷ √6.
2. Rationalize 5 ÷ √2.
3. Rationalize 3 ÷ (2√5).
4. Rationalize 1 ÷ (3 + √2) using a conjugate.
5. Rationalize 1 ÷ cube root of 5.
6. Explain why you multiply the numerator and denominator by the same expression.

Answer Section

Final Takeaway

To rationalize a denominator, multiply by a form of 1 that creates a rational number on the bottom. Use the same radical for a single-root denominator and a conjugate for a binomial denominator. Then simplify carefully.