Rationalize Denominator Calculator
Choose the type of expression you want to rationalize, enter the coefficients and radicands, and the calculator returns the fully simplified rational form with every step shown. Four modes cover the most common cases: a simple monomial square root in the denominator, a binomial denominator requiring the conjugate method, a cube root in the denominator, and a general nth root.
What is rationalizing the denominator?
Rationalizing the denominator is the process of rewriting a fraction so that no radicals (square roots, cube roots, or other roots) appear in the denominator. The value of the fraction does not change - the numerator and denominator are multiplied by the same expression, which equals 1. The result is an equivalent fraction that is easier to work with in further calculations, simpler to compare with other fractions, and consistent with the standard form required in most textbooks and exams.
The three main rationalization techniques
For a monomial denominator containing a square root, such as a / (b*sqrt(c)), multiply both the numerator and denominator by sqrt(c). The denominator becomes b * sqrt(c) * sqrt(c) = b * c, which is rational. For a binomial denominator, such as a / (p + q*sqrt(r)), the conjugate method is used: multiply top and bottom by (p - q*sqrt(r)). The denominator becomes (p + q*sqrt(r))(p - q*sqrt(r)) = p^2 - q^2*r using the difference-of-squares identity. For a cube root in the denominator, a / (b*cbrt(c)), multiply by cbrt(c^2)/cbrt(c^2). The denominator becomes cbrt(c) * cbrt(c^2) = cbrt(c^3) = c.
Why radicals in the denominator are avoided
Historically, computing a value like 1/sqrt(2) required dividing by an irrational number, which is harder by hand than dividing by an integer. By rewriting as sqrt(2)/2, the division is straightforward. Today calculators make the arithmetic easy either way, but the convention persists because it gives a canonical, simplified form: every expression has exactly one rationalized form, making answers easy to check, compare, and grade. In algebra and calculus courses, an answer with a radical in the denominator is typically considered unsimplified.
How to verify your result
The fastest check is to evaluate both the original fraction and the rationalized result numerically using a calculator. For example, 3/(2*sqrt(5)) gives approximately 0.6708. The rationalized form 3*sqrt(5)/10 also gives 3*2.2361/10 = 6.7082/10 = 0.6708. The values match, confirming no error. The decimal approximation field in this calculator performs exactly this check for you automatically.
Rationalization methods at a glance
| Denominator type | Example | Rationalizing factor | Result denominator |
|---|---|---|---|
| Monomial: b*sqrt(c) | 3 / (2*sqrt(5)) | sqrt(5) / sqrt(5) | 2 * 5 = 10 |
| Binomial: p + q*sqrt(r) | 1 / (2 + sqrt(3)) | (2 - sqrt(3)) / (2 - sqrt(3)) | p^2 - q^2*r = 4 - 3 = 1 |
| Binomial: p - q*sqrt(r) | 1 / (2 - sqrt(3)) | (2 + sqrt(3)) / (2 + sqrt(3)) | p^2 - q^2*r = 4 - 3 = 1 |
| Cube root: b*cbrt(c) | 5 / cbrt(4) | cbrt(16) / cbrt(16) | b * c = 4 |
| nth root: b * n-rt(c) | 2 / (n-rt(3)), n=4 | 4-rt(27) / 4-rt(27) | b * c = 3 |
Choose the method that matches the structure of your denominator.
Frequently asked questions
Why do we rationalize the denominator?
Rationalizing the denominator is a convention that puts fractions into a standard simplified form. An expression with a radical in the denominator is not considered fully simplified in most algebra and calculus courses. Historically it also made hand calculations easier: dividing by an integer is simpler than dividing by an irrational number. The value of the fraction does not change - only its form.
What is the conjugate method and when is it used?
The conjugate of a binomial a + b is a - b (you flip the sign of the second term). When a denominator is of the form p + q*sqrt(r), multiplying top and bottom by the conjugate (p - q*sqrt(r)) exploits the difference-of-squares identity (p + q*sqrt(r))(p - q*sqrt(r)) = p^2 - q^2*r. Because p^2 and q^2*r are both rational, their difference is rational. This method is used whenever the denominator is a two-term expression containing a radical.
How do you rationalize a cube root in the denominator?
For a denominator of the form b*cbrt(c), multiply top and bottom by cbrt(c^2). The denominator becomes b * cbrt(c) * cbrt(c^2) = b * cbrt(c^3) = b * c, which is a rational integer. The numerator gains a factor of cbrt(c^2), but it now sits in the numerator where a radical is acceptable. The same principle generalizes to any nth root: multiply by the (n-1)th power of the radicand inside the root.
Does rationalizing the denominator change the value of the expression?
No. You are multiplying the fraction by a form of 1 (something divided by itself), so the value is preserved. This is the same principle as multiplying 1/2 by 2/2 to get 2/4 - both equal 0.5. The decimal approximation in this calculator confirms the original and rationalized expressions are numerically equal.
What if the denominator has more than two radical terms?
Multi-term radical denominators like a + sqrt(b) + sqrt(c) require repeated application of the conjugate method. You first rationalize one pair of terms to get a simpler expression, then rationalize again. This can become quite involved and is typically encountered in advanced algebra or competition mathematics. The four modes in this calculator handle the most common cases encountered at the high-school and early college level.
Can I rationalize a denominator that contains a variable under the radical?
Yes, the process is the same. For 1/sqrt(x), multiply by sqrt(x)/sqrt(x) to get sqrt(x)/x (assuming x is positive). For 1/(a + sqrt(x)), multiply by (a - sqrt(x))/(a - sqrt(x)) to get (a - sqrt(x))/(a^2 - x). This calculator works with integer radicands for clarity of the step-by-step output, but the identical technique applies when the radicand is a variable.