A rationalize the denominator calculator rewrites a fraction so that no square root remains in the denominator. It does not change the value of the fraction; it changes its form to a standard form that is easier to compare, add, and use in further algebra. Textbook answers are usually expected in rationalized form, so a fraction with a root in the denominator is often marked incomplete even when its value is correct.
The method uses a simple trick: multiply numerator and denominator by the same root, which is equivalent to multiplying by 1 in a disguised form. Because the denominator becomes a root times itself, it becomes a whole number. For 1 / sqrt(2), multiplying top and bottom by sqrt(2) gives sqrt(2) / 2, which has the same value, about 0.707, but no root in the denominator.
What this calculator does
This guide covers the case a / sqrt(b), where a is a whole number numerator and b is the number under the root. The rationalized form is (a x sqrt(b)) / b. The calculator returns that expression and its decimal value, so you can see both the exact form and the approximate size. For a denominator with a sum or difference, such as 1 / (2 + sqrt(3)), the multiplier is the conjugate, 2 – sqrt(3), which is a different but related method and is not covered by this calculator.
Rationalizing is a convention, not a law of nature. A calculator that returns 0.707 for 1 / sqrt(2) is giving the same number as sqrt(2) / 2, just in decimal form. The rationalized form is preferred in exact algebra because it keeps the root visible and avoids a repeating decimal approximation in the denominator.
Formula
For a / sqrt(b): rationalized form = (a x sqrt(b)) / b, provided b is positive and not zero.
How to use it
- Identify a, the numerator, and b, the number under the square root in the denominator.
- Multiply numerator and denominator by sqrt(b).
- Simplify the denominator: sqrt(b) x sqrt(b) = b.
- Write the result as (a x sqrt(b)) / b, and simplify the fraction if a and b share a common factor.
- Check by decimal: the original and rationalized forms must give the same decimal value.
Try the calculator
Worked example
Rationalize 1 / sqrt(2). Here a = 1 and b = 2. Multiply top and bottom by sqrt(2) to get sqrt(2) / 2. As a decimal, sqrt(2) is about 1.414, so sqrt(2) / 2 is about 0.707. The original, 1 / 1.414, is also about 0.707, confirming the rewrite. The rationalized answer is therefore sqrt(2) / 2, not 1 / sqrt(2).
A second example: 3 / sqrt(5) becomes (3 x sqrt(5)) / 5. Numerically, sqrt(5) is about 2.236, so 3 x 2.236 = 6.708, divided by 5 = 1.342. The original, 3 / 2.236, is also about 1.342, confirming the method.
How to check your answer
Compute the decimal value of both the original and the rationalized form. They must match.
Check that no square root remains in the denominator of the final answer.
Simplify any common factor between the numerator coefficient and the denominator after rationalizing.
Test the rationalized form with a concrete value, such as a = 2 and b = 3, where the original is about 1.155 and the rationalized form, 2 times root 3 over 3, must give the same decimal. A mismatch at this stage means the multiplier was applied to only one part of the fraction.
Common mistakes
- Multiplying only the denominator by the root and leaving the numerator unchanged, which changes the value.
- Writing sqrt(b) x sqrt(b) as sqrt(2b) or as 2b instead of b.
- Rationalizing with the wrong multiplier for a binomial denominator, where the conjugate is needed instead of the root alone.
- Leaving a root in the denominator and treating the fraction as fully simplified.
Frequently asked questions
Why rationalize a denominator at all?
It puts fractions in a standard exact form that is easier to compare and to use in further algebra, and it is the form textbooks usually expect.
Does rationalizing change the value?
No. Multiplying top and bottom by the same non-zero number does not change the value, so the rationalized form and the original form are equal.
What about a denominator like 2 + sqrt(3)?
Multiply top and bottom by the conjugate, 2 - sqrt(3). The denominator becomes 4 - 3 = 1, with no root left, which is the same idea with a different multiplier.
