About the Rationalize the Denominator Calculator
This rationalize the denominator calculator rewrites a fraction with square roots in the denominator as an equivalent fraction whose denominator is a whole number. It handles a single radical such as 1/√2 or 6/√12, and binomial denominators such as 4/(3 − √5), by multiplying the top and bottom by the right factor — √u for a lone radical, or the conjugate s − t√u for a sum.
Enter the numerator as p + q√r and the denominator as s + t√u using whole numbers (set q or t to 0 if there is no radical part). The calculator simplifies every radical first (√12 = 2√3), multiplies out, collects like radicals, cancels common factors and shows each step, plus a decimal check that the value has not changed.
It is meant for algebra and precalculus students practising simplest radical form, teachers making answer keys, and anyone who needs an exact answer in the conventional form textbooks and exams expect. Only square roots are supported; cube-root denominators need a different multiplier (a² ∓ ab + b²).
How to use the rationalize the denominator calculator
- 1Write the numerator as p + q√r and enter p, q and r (q = 0 if there is no radical).
- 2Write the denominator as s + t√u and enter s, t and u — for 3 − √5 use s = 3, t = −1, u = 5.
- 3For a lone radical like 1/√2, set s = 0, t = 1 and u = 2.
- 4Read the rationalized form and follow the steps table.
- 5Compare the decimal value to confirm the answer equals the original fraction.
Formula and method
Multiplying a fraction by a form of 1 does not change its value. For a single radical in the denominator, multiply by √u/√u, because √u·√u = u. For a binomial denominator s + t√u, multiply by its conjugate s − t√u: the difference-of-squares identity (s + t√u)(s − t√u) = s² − t²u removes the radical entirely.
Before multiplying, each radical is put in simplest form by pulling out square factors (√12 = 2√3), so the conjugate is as small as possible. After multiplying, products of radicals are simplified again (√2·√6 = 2√3), like radicals are collected, and any factor common to every coefficient and the denominator is cancelled. A negative denominator is made positive by moving the sign to the top.
- p + q√r
- Numerator (whole numbers p, q, r ≥ 0)
- s + t√u
- Denominator (whole numbers s, t, u ≥ 0)
- s − t√u
- Conjugate of the denominator
- s² − t²u
- Rational denominator after multiplying by the conjugate
Worked examples
4 / (3 − √5)
Multiply by the conjugate 3 + √5. The denominator becomes 3² − 5 = 4 and the numerator 4(3 + √5) = 12 + 4√5. Dividing everything by 4 gives 3 + √5 ≈ 5.2361.
Single radical: 1 / √2
Multiply top and bottom by √2: the numerator becomes √2 and the denominator √2·√2 = 2, so 1/√2 = √2/2 ≈ 0.7071.
Radicals on top and bottom: (1 + √2) / (3 − √2)
The conjugate is 3 + √2. The denominator is 9 − 2 = 7. The numerator is (1 + √2)(3 + √2) = 3 + √2 + 3√2 + 2 = 5 + 4√2, giving (5 + 4√2)/7 ≈ 1.5224.
Simplify first: 6 / √12
√12 = 2√3, so the fraction is 6/(2√3). Multiplying by √3/√3 gives 6√3/6, and cancelling 6 leaves √3 ≈ 1.7321.
Frequently asked questions
What does it mean to rationalize the denominator?+
It means rewriting a fraction so that no radical (such as a square root) remains in the denominator, without changing its value. For example, 1/√2 becomes √2/2. The result is considered simplest radical form.
How do you rationalize a denominator with two terms?+
Multiply the numerator and denominator by the conjugate — the same two terms with the sign between them flipped. Because (a + b)(a − b) = a² − b², the square root disappears from the denominator.
Why do we rationalize denominators?+
Historically it made hand division easier (dividing by 2 is simpler than by 1.41421…), and today it gives a standard form so answers can be compared. Many textbooks and exams require answers with rational denominators.
What is the conjugate of 3 − √5?+
The conjugate is 3 + √5. Multiplying them gives 3² − (√5)² = 9 − 5 = 4, a whole number, which is exactly why conjugates are used to rationalize.
Can I rationalize a cube root denominator?+
Yes, but not with a conjugate. For 1/∛a multiply by ∛a²/∛a²; for a − ∛b use the sum-of-cubes factor a² + a∛b + ∛b². This calculator handles square roots only.
Does rationalizing change the value of the fraction?+
No. You multiply by a fraction equal to 1, such as √2/√2 or (3 + √5)/(3 + √5), so the value is unchanged. The decimal check shown with each answer confirms this.