Enter a fraction with a square root, cube root or a two-term radical in the denominator, and get the exact answer with no root left on the bottom. Every step is shown: simplifying the radicals, choosing the multiplier or conjugate, expanding and reducing.
Last updated:
Denominator type
Use 0 to leave out this term.
Negative for a minus sign, e.g. 3 − √2 → d = −1.
1 means no root in the numerator.
Enter your values to see the result.
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How to use the Rationalize the Denominator Calculator
Pick the type: a binomial denominator (two terms) or a single root such as √12 or ∛4.
Enter the coefficients and the numbers under each root. Use 1 under the root for a plain number and 0 to drop a term.
Use negative coefficients for minus signs, for example 3 − √2 is c = 3 and d = −1.
Press Rationalize to see the exact answer, the multiplier used and the decimal value.
Read the steps to see how the denominator becomes a whole number.
How rationalizing works
A fraction keeps its value when you multiply the top and bottom by the same nonzero number. Rationalizing picks that number so the denominator turns into a whole number:
One square root, √c: multiply by √c, because √c · √c = c.
Cube root or higher, ⁿ√c: multiply by the root that completes a perfect n-th power. After simplifying, if c = pe·qf… multiply by ⁿ√(pn−e·qn−f…).
Two terms, a + b√q or a√p + b√q: multiply by the conjugate (same terms, opposite middle sign). The difference of squares (x + y)(x − y) = x² − y² removes every square root.
Always simplify the radicals first (√12 = 2√3, ∛16 = 2∛2). That keeps the multiplier small and the final answer easy to reduce.
Worked examples
Expression
Multiply top and bottom by
New denominator
Simplified result
1/√2
√2
2
√2/2
6/√12 = 6/(2√3)
√3
6
√3
√3/√8 = √3/(2√2)
√2
4
√6/4
5/(2∛3)
∛9
6
5∛9/6
1/∛4
∛2
2
∛2/2
4/(3 + √5)
3 − √5
9 − 5 = 4
3 − √5
(1 + √2)/(3 − √2)
3 + √2
9 − 2 = 7
(5 + 4√2)/7
1/(√3 − √2)
√3 + √2
3 − 2 = 1
√3 + √2
(2 + √3)/(2 − √3)
2 + √3
4 − 3 = 1
7 + 4√3
Tips for checking your work
Compare decimals: the original and the rationalized form must give the same value (the calculator shows it).
Watch the sign of the conjugate. For 3 − √2 the conjugate is 3 + √2, not −3 − √2.
Cube-root binomials such as 1/(1 + ∛2) need the sum-of-cubes identity a³ + b³ = (a + b)(a² − ab + b²), so the multiplier is 1 − ∛2 + ∛4. Those are outside this calculator's two modes.
If the denominator simplifies to a whole number (for example √9 = 3), there is nothing left to rationalize; just reduce the fraction.
Frequently asked questions
What does it mean to rationalize the denominator?
It means rewriting a fraction so that the denominator contains no radicals (square roots, cube roots and so on). The value stays the same because you multiply the top and bottom by the same nonzero expression. For example 1/√2 becomes √2/2.
How do you rationalize a denominator with a square root?
Multiply the numerator and denominator by the root in the denominator. Since √c × √c = c, the denominator becomes a whole number. Simplify the radical first: 6/√12 = 6/(2√3), then multiplying by √3 gives 6√3/6 = √3.
How do you rationalize a cube root in the denominator?
Multiply by the cube root that completes a perfect cube. For 5/(2∛3) you need two more factors of 3, so multiply by ∛9: the denominator becomes 2∛27 = 6 and the answer is 5∛9/6. In general for ⁿ√c you multiply by ⁿ√(c^(n−1)), using the smallest exponents after simplifying.
What is a conjugate and why does it work?
The conjugate of a + √b is a − √b. Multiplying them uses the difference of squares, (a + √b)(a − √b) = a² − b, which has no root. So for 4/(3 + √5) you multiply by (3 − √5)/(3 − √5) to get 4(3 − √5)/4 = 3 − √5.
Can I rationalize a denominator with two square roots like √3 − √2?
Yes. The conjugate is √3 + √2, and (√3 − √2)(√3 + √2) = 3 − 2 = 1, so 1/(√3 − √2) = √3 + √2. Enter c = 1, p = 3, d = −1, q = 2 in the binomial mode.
Is an answer with a root in the denominator wrong?
Not mathematically: 1/√2 and √2/2 are equal. Rationalized form is the conventional simplest form in algebra classes and makes fractions easier to compare and add, so many teachers and tests expect it.
Why are my answers reduced at the end?
After rationalizing, the numerator terms and the new whole-number denominator often share a common factor. Dividing everything by their greatest common divisor gives the fully simplified result, for example 6√3/6 reduces to √3.
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