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Rationalize the Denominator Calculator

Enter a numerator and a radicand to rationalize a/√b.

Result

2.5√12 / 12
2√3
0.7217

How it is calculated

For a / √b the tool multiplies top and bottom by √b: a / √b = a√b / b, because √b × √b = b. The first output prints exactly that, with b as originally typed, and it does not cancel the fraction (6√3 / 3 is not reduced to 2√3). The second output is the root √b alone, simplified by pulling out the largest square factor: √b = k√m where k² is the biggest perfect square dividing b. The decimal output is a ÷ √b to four places, calculated from the exact numerator. The numerator may be any number, including a decimal or negative one, and in the first line it is printed with up to four decimals (trailing zeros dropped), so 2.567 appears as 2.567 and 2.50 as 2.5. The radicand must be a positive whole number up to one trillion, so 0, negatives, values like 2.5 and larger numbers give no result.

Worked example

Rationalize 2.5 / √12. Multiply top and bottom by √12: numerator 2.5√12, denominator 12, so the form is 2.5√12 / 12. Simplify the root: 12 = 4 × 3, so √12 = 2√3, which is the second output. Then the expression is 2.5 × 2√3 ÷ 12 = 5√3 ÷ 12. Decimal check: 2.5 ÷ 3.4641 = 0.7217, the same as 5 × 1.7321 ÷ 12.

Try this example →

Common mistakes

  • Expecting the first output to be fully reduced: it keeps b as typed, so you still have to cancel common factors yourself, as 6√3 / 3 = 2√3.
  • Reading the simplified-radical line as the whole answer: it is only √b simplified (for example 2√3), and the numerator a and the denominator are not included.
  • Entering the radicand as a decimal such as 2.5, or a negative number: only positive whole numbers work, and anything else gives no result.
  • Putting the whole denominator in the radicand when it is something like 3 + √2: this tool handles only a single square root a/√b, not a binomial denominator.
  • Treating a perfect square, for example b = 16, as irrational: the radical comes back as the plain whole number 4, and no square root remains.

About this calculator

Rationalizing the denominator means rewriting a fraction like a/√b so that no square root remains on the bottom, by multiplying both the numerator and denominator by √b. Since √b × √b = b, the denominator becomes the rational number b, and the fraction becomes a√b / b — mathematically identical to the original, just written in a different, standardized form.

The convention exists because a rationalized denominator is easier to compare, add, and simplify: two fractions with irrational denominators are hard to combine directly, but once both have been rationalized, combining them is ordinary fraction arithmetic. It is a staple of algebra and pre-calculus courses, where students are expected to leave radical answers in rationalized form, and it also matters historically — before calculators, dividing by a rational number by hand was far easier than dividing by an irrational decimal approximation.

Enter a numerator a and a radicand b, and the calculator shows a/√b as a√b / b (not reduced any further, so 6√3 / 3 stays as written), the simplified form of √b on its own, and the decimal value. You can reduce the fraction yourself by cancelling common factors.

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