Math
Rationalize the Denominator Calculator
Pick the shape of your fraction, enter the numbers, and get the denominator rationalized step by step.
Simple radicals are cleared by multiplying by the root, cube roots by the matching power, and binomial denominators by their conjugate. Every answer is simplified, reduced, and checked against the decimal value of the original fraction.
A whole number. Negatives are fine.
The number under the root in the denominator.
Try an example
Result
Rationalized form
—
- Original fraction
- —
- Multiply top and bottom by
- —
- Decimal check
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the rationalize the denominator calculator solves
Pick the shape of your fraction, enter the numbers, and get the The bottom number of a fraction; it says how many equal parts make one whole. rationalized step by step. Simple A root expression written with the √ symbol, such as a square root. are cleared by multiplying by the root, cube roots by the matching power, and binomial denominators by their conjugate. Every answer is simplified, reduced, and checked against the decimal value of the original fraction.
Formula
How to use it
- Choose the denominator type that matches your fraction.
- Enter the The top number of a fraction; it counts how many parts you have. a and the radicand b; for binomial denominators also enter the sign and the second term c.
- Read the rationalized form, the multiplier or conjugate used, and the decimal check showing both forms agree.
How to read the answer
Rationalizing does not change the value of the fraction, only how it is written: the decimal check confirms the original and the answer are equal. For a single radical, multiplying by the same radical (or the matching power for cube roots) clears it. For a sum or difference, the conjugate turns the denominator into a difference of squares, which has no radical left.
Common mistakes and edge cases
- Multiplying only the denominator. Whatever multiplies the bottom must multiply the top too.
- Using √b for a cube root: ∛2 × ∛2 = ∛4, still a radical. You need ∛4 so that ∛2 × ∛4 = ∛8 = 2.
- Squaring the conjugate instead of multiplying the two binomials: (√3 + 1)(√3 - 1) = 3 - 1 = 2, not (√3 + 1)².
- Forgetting to reduce: 6√2/4 simplifies to 3√2/2.
- Choosing c so that √b = c, which makes the original denominator 0.
Worked examples
3/√2
Multiply by √2/√2 to get 3√2/2.
Rationalized form
3√2/2
6/√8 with simplification
√8 = 2√2, so 6/(2√2) = 3/√2 = 3√2/2.
Rationalized form
3√2/2
Cube root 5/∛2
Multiply by ∛4/∛4 to get 5∛4/2.
Rationalized form
5∛4/2
Conjugate 2/(√3 + 1)
Multiply by (√3 - 1): 2(√3 - 1)/2 = √3 - 1.
Rationalized form
√3 - 1
Two radicals 4/(√7 - √3)
Denominator becomes 7 - 3 = 4.
Rationalized form
√7 + √3
Perfect square 5/√9
Already rational: 5/3.
Rationalized form
5/3
Zero denominator 1/(√4 - 2)
Undefined because √4 = 2.
Rationalized form
Error
Frequently asked questions
What does it mean to rationalize the denominator?+
Rewrite the fraction so no radical (irrational number) remains in the denominator, without changing its value. 3/√2 becomes 3√2/2.
How do I rationalize a denominator with a square root?+
Multiply the numerator and denominator by that square root: a/√b × √b/√b = a√b/b. Simplify the radical and reduce the fraction afterward.
How do I rationalize a cube root denominator?+
Multiply by whatever power completes a perfect cube under the radical. For ∛2 that is ∛4, since ∛2 × ∛4 = ∛8 = 2, giving 5/∛2 = 5∛4/2.
What is a conjugate?+
The same two terms with the middle sign flipped: the conjugate of √3 + 1 is √3 - 1. Multiplying a binomial by its conjugate gives a difference of squares, which removes the radical.
Why do teachers ask for rationalized denominators?+
It gives a standard form so answers can be compared, and historically it made hand division easier: dividing by 2 is simpler than dividing by 1.41421…
Does rationalizing change the value?+
No. You multiply by a fraction equal to 1, so the value is identical. The decimal check on this page shows both forms agree.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Simplify Radicals Calculator
Simplify √, ∛, ∜, and fifth roots to simplest radical form using a prime-factor table, with coefficients (5√72 = 30√2), negative radicands, the decimal value, and the rational-exponent form.
Square Root Calculator
Calculate square roots, simplify exact radicals for whole numbers, and identify perfect squares.
Cube Root Calculator
Find the cube root of any number with an exact answer for perfect cubes, simplified radical form (∛54 = 3∛2), the decimal value, nearest perfect cubes, and full support for negative and decimal inputs.
Fraction Calculator
Add, subtract, multiply, divide, simplify, compare, or convert decimals to fractions with steps and visual fraction bars.
Nth Root Calculator
Find the nth root of a number for any index: exact simplified radical form, decimal value, the rational-exponent form x^(m/n), and fractional powers such as 81^(3/4) = 27, with steps.
Adding Fractions Calculator
Add up to five fractions or mixed numbers. Finds the least common denominator with prime factors, rewrites each fraction, adds, simplifies, and shows the mixed-number and decimal forms with a fraction-strip picture.
Last updated: September 4, 2026