Rationalizing a denominator means rewriting the fraction so that no root is left on the bottom. The value does not change: 1/(√3 − 1) and (√3 + 1)/2 are the same number, ≈ 1.366. You multiply top and bottom by a factor equal to 1, and the factor depends on what is in the denominator.
| Denominator | Multiply by | Example |
|---|---|---|
| A root, √c | √c/√c | 10/√6 = 5√6/3 |
| A number times a root, b√c | √c/√c | 3/(2√5) = 3√5/10 |
| Binomial with a square root | The conjugate over itself | 1/(√3 − 1) = (√3 + 1)/2 |
| Cube root, ∛a | ∛(a²)/∛(a²) | 1/∛2 = ∛4/2 |
Read the table from the left column: first look at what is on the bottom, then choose the factor. The conjugate of a binomial is the same binomial with the sign in the middle changed.
Worked example: 1/(√3 − 1)
Rationalize the denominator of 1/(√3 − 1).
- Write the conjugate. The denominator is √3 − 1, so the conjugate is √3 + 1.
- Multiply top and bottom by (√3 + 1)/(√3 + 1), which equals 1.
- Denominator. (√3 − 1)(√3 + 1) = 3 − 1 = 2, because the middle terms cancel.
- Numerator. 1 × (√3 + 1) = √3 + 1.
- Put it together. The result is (√3 + 1)/2.
Result: (√3 + 1)/2.
Check: (√3 + 1)/2 × (√3 − 1) = (3 − 1)/2 = 1, the starting numerator. With decimals, 1 ÷ 0.7321 and 2.7321 ÷ 2 both give ≈ 1.366.
The trick in step 3 is the difference of squares: (x − y)(x + y) = x² − y². If x or y is a square root, its square is a whole number and the radical disappears.
A lone root, with or without a number in front
When the bottom is b√c, multiply by √c/√c. Since √c × √c = c, the denominator becomes b × c. The rule is a/(b√c) = a√c/(bc), with b not equal to 0 and c positive.
For 3/(2√5) you get (3√5)/(2 × 5) = 3√5/10, which is ≈ 0.6708. For 10/√6 you get 10√6/6, and because 10 and 6 share the factor 2, it reduces to 5√6/3. To review how to reduce, the simplify fractions tool does it for any pair of integers.
If the root can be simplified, do that first. In 1/√8, since √8 = 2√2, you work with 1/(2√2) = √2/4 and avoid large numbers.
A binomial with a root: the conjugate
Multiplying by the same root does not work when the bottom is a sum or a difference. In (√3 − 1) × √3 = 3 − √3 the root is still there. The conjugate does remove it, because it turns the denominator into a difference of squares.
With 5/(√6 + 1), the conjugate is √6 − 1. The denominator becomes 6 − 1 = 5 and the numerator 5(√6 − 1), so the 5 cancels and the result is √6 − 1, which is ≈ 1.449. When both terms have a root, as in 6/(√5 − √2), the denominator is 5 − 2 = 3 and the result is 2(√5 + √2).
A cube root: completing the cube
With ∛a, multiplying by ∛a gives ∛(a²), which is still a root. You need the factor that completes a perfect cube: ∛a × ∛(a²) = ∛(a³) = a.
For 1/∛2, multiply by ∛4/∛4. The bottom becomes ∛2 × ∛4 = ∛8 = 2, and the result is ∛4/2 ≈ 0.794. For 6/∛9 the factor is ∛3, because ∛9 × ∛3 = ∛27 = 3, and you get 6∛3/3 = 2∛3. The typical error is to multiply by ∛2 in the first case: the bottom comes out as ∛4 and the root does not disappear.
Practice
Four exercises
- Rationalize 9/√3.
- Rationalize 4/(3√2).
- Rationalize 6/(√5 − √2).
- Rationalize 1/∛3.
Show the answers
- 3√3 (you get 9√3/3 and reduce)
- 2√2/3 (4√2/6 reduces by dividing by 2)
- 2√5 + 2√2 (the conjugate gives a denominator of 5 − 2 = 3, and 6/3 = 2)
- ∛9/3 (∛3 × ∛9 = ∛27 = 3)
Where to go next
To compute a root of a fraction and see its decimal, use the fraction roots calculator. If it is hard to find the line where you went wrong, the guide to fraction mistakes collects the most common slips, and the worked fraction exercises give more practice. With letters instead of numbers, domain restrictions are explained in algebraic fractions and domain. Roots as fractional exponents are covered in negative and fractional exponents, and products are reviewed in multiply fractions.
Sources
English Wikipedia, article “Rationalisation (mathematics)”: when the denominator is a binomial with a square root, you multiply the numerator and the denominator by the conjugate of the denominator, and the difference of squares removes the root (accessed October 17, 2026).