Rationalize the Denominator Calculator
1/√2 and √2/2 are the same number. Work them out and both give 0.7071. But one of them is the answer your teacher wants, and the other is not.
Moving the radical off the bottom is what this rationalize the denominator calculator does — and it shows what the fraction was multiplied by, which is the part worth learning.
What Does It Mean to Rationalize the Denominator?
What does rationalize the denominator mean in practice? It means rewriting a fraction so that nothing under a root sign is left in the bottom. The value stays exactly the same — only the shape changes.
The reason is older than it looks. Before calculators, working out 1 ÷ 1.41421 by hand was miserable, while 1.41421 ÷ 2 was easy. So people moved the radical upstairs, and the habit outlived the reason. Today it is convention, and it is still marked.
There is one practical benefit left. Two answers written differently are hard to compare, but every rationalized answer is written the same way — which is exactly what a marking scheme needs.
How to Rationalize the Denominator
There are two shapes, and knowing which you are looking at is most of the work.
One radical underneath. Multiply top and bottom by that radical. 1/√2 × √2/√2 = √2/2, because √2 × √2 is just 2. You are multiplying by 1, so the value cannot change.
Two terms underneath. Multiplying by the radical alone does not work here — it leaves another radical behind. You need the conjugate: the same two terms with the sign between them flipped.
1/(2+√3) becomes 1/(2+√3) × (2−√3)/(2−√3). The bottom is now (2+√3)(2−√3) = 4 − 3 = 1, so the whole answer is simply 2 − √3.
That works because of one identity: (a+b)(a−b) = a² − b². Squaring is precisely what kills a square root, and the conjugate squares both terms in a single step.
How to Use This Rationalize the Denominator Calculator
Type the fraction and press =. The keypad has √( and ∛( , brackets and the digits, so 1/(2+√3) is a handful of taps.
Worksheets usually ask you to rationalize the denominator and simplify in one go, and the order matters. Do the simplifying first — it makes everything shorter.
Simplifying by rationalizing the denominator is really two habits at once, and this is where they meet. 3/√12 looks awkward until you notice √12 = 2√3. Then it is 3/(2√3), which clears to √3/2 in one move. Going straight at √12 works too, but you end up reducing a much uglier fraction at the end.
A cube root needs a little care as well. 1/∛2 does not clear by multiplying by ∛2 — that gives ∛4, which is still a root. You need ∛4, so that ∛2 × ∛4 = ∛8 = 2. The rule is to complete the whole power, whatever the index is.
This calculator does both automatically: it simplifies what is under the sign before it clears, and works out how much of a root is needed to complete the power. If you only want the radical itself tidied up, the simplify radicals calculator does that half on its own.
Rationalizing the Denominator FAQ
What does it mean to rationalize the denominator?
How do you rationalize the denominator?
What is the conjugate?
Why do we rationalize the denominator at all?
Do you rationalize the denominator and simplify at the same time?
One radical, multiply by it. Two terms, multiply by the conjugate. Simplify first and the numbers stay small — that is the whole topic.