The rationalize the denominator calculator above takes a fraction whose bottom contains a square root and rewrites it so the bottom is a whole number. Where the denominator is a single radical it multiplies top and bottom by that radical. Where the denominator is a binomial such as 3 + √5 it uses the conjugate, 3 − √5, which turns the bottom into a difference of two squares and removes the surd in one step. The answer comes back in simplest exact form, with the radical simplified and the whole fraction reduced.
Arb Digital publishes this page because the technique is mechanical but the reason for it is rarely explained, and students who do not know why they are doing it make predictable mistakes — multiplying by the wrong thing, or multiplying only the bottom. The tool shows the multiplier it used and the denominator that came out, so the working is visible rather than assumed, and it prints the decimal value of both the original and the result as a self-check.
What This Calculator Does
It handles fractions of the form (p + q√r) ÷ (s + t√u), where the surd in the numerator, if there is one, simplifies to the same irrational square root as the one in the denominator. That covers essentially every rationalizing exercise in a school algebra course: a whole number over a surd, a surd over a surd, and a binomial denominator with or without a surd on top. It simplifies the radicands first, so entering 12 is treated as 2√3 throughout, and it reduces the final fraction by the greatest common divisor of all three parts.
This is a different job from our square root calculator, which simplifies a single radical into the form k√m and gives its decimal value. That page tells you √72 is 6√2; this page takes 1 ÷ (4 + √72) and turns it into a fraction with no radical underneath. The two are complementary, and this tool calls on the same simplification internally before it does anything else.
How to Use It
- Choose the denominator shape. Binomial for anything of the form s + t√u; single radical when there is no whole-number term.
- Enter the numerator. For a plain number, put it in p and leave q at zero. For something like 2 + 3√5, set p = 2, q = 3 and r = 5.
- Enter the denominator the same way. Use a negative t for a minus sign, so 3 − √5 is s = 3, t = −1, u = 5.
- Read the multiplier. That is the expression you would write on both the top and the bottom if you were doing it by hand.
- Compare the two decimals. They must match. If they do not, an input was mistyped.
The Method and How It's Calculated
For a single radical denominator the idea is simply that √u × √u = u. Multiplying top and bottom by √u leaves the value unchanged, because you have multiplied by 1, but the bottom becomes rational. So 3 ÷ √7 becomes 3√7 ÷ 7.
For a binomial denominator that trick fails, because (s + t√u)² still contains a cross term with a surd in it. The conjugate solves it. The conjugate of s + t√u is s − t√u, and their product is the difference of two squares: (s + t√u)(s − t√u) = s² − t²u. The cross terms cancel exactly, and since s, t and u are all whole numbers the result is a whole number.
Take the defaults: 5 ÷ (3 + √5). Multiply top and bottom by the conjugate 3 − √5. The bottom becomes 3² − 1²×5 = 9 − 5 = 4. The top becomes 5(3 − √5) = 15 − 5√5. So the answer is (15 − 5√5) ÷ 4. Check it numerically: √5 is about 2.2360680, so the original is 5 ÷ 5.2360680 = 0.9549150, and the answer is (15 − 11.1803399) ÷ 4 = 3.8196601 ÷ 4 = 0.9549150. Identical, as it must be. Paul's Online Notes covers the same manipulations in its section on Radicals.
Why Bother Rationalizing At All?
The historical reason is obsolete and worth knowing anyway. Before calculators, dividing by a long decimal by hand was laborious and dividing by a whole number was easy. Working out 1 ÷ 1.4142136 required real effort; working out 1.4142136 ÷ 2 took seconds. Rationalizing converted a hard division into an easy one, which made it a genuine labour-saving step rather than a convention.
Three reasons survive. The first is comparison: two expressions written with rational denominators can be compared and combined at a glance, while √2 ÷ 2 and 1 ÷ √2 look completely different despite being the same number. The second is standard form — examiners and textbooks expect a rationalized answer, and an unrationalized one may be marked incomplete regardless of whether it is correct. The third is numerical: in floating-point arithmetic, dividing by a small irrational quantity can amplify rounding error, and rearranging so the division is by a well-behaved integer is a standard numerical-stability move.
There is a fourth that matters more than the others in practice. Rationalizing often reveals structure that was hidden. When you rationalize 1 ÷ (√(x+h) − √x), the numerator turns into h and the whole expression collapses — that manipulation is the key step in differentiating a square root from first principles, and nobody would find it by looking at the original form. The technique is not really about tidiness; it is about turning a subtraction of surds into something that cancels.
Simplify the Radical First
Always reduce the radicand before rationalizing, because doing it afterwards leaves you with a larger fraction to reduce and more chances to slip. √12 is 2√3, √50 is 5√2, √72 is 6√2. The rule is to pull out the largest perfect square factor: 12 = 4 × 3, and √4 is 2.
This matters more than it looks with binomial denominators. The expression 1 ÷ (1 + √12) has the conjugate 1 − √12 and produces a denominator of 1 − 12 = −11. Simplify first and it is 1 ÷ (1 + 2√3), with conjugate 1 − 2√3 and denominator 1 − 4×3 = −11 — the same, as it must be, but the intermediate numbers are smaller and the sign is easier to keep track of. The calculator does this simplification automatically, which is why entering u = 12 and entering u = 3 with t = 2 give identical answers. Our prime factorization calculator is the quickest way to spot the square factors in a large radicand by hand.
What Happens When the Denominator Goes Negative
Nothing goes wrong, but the sign needs care. If t²u is larger than s², the rationalized denominator s² − t²u comes out negative. The standard convention is to keep the denominator positive, so both the numerator and the denominator are multiplied by −1 and the signs on the top flip. The calculator does this automatically, which is why an answer can appear with a leading minus on a term that was positive in your working.
The genuinely bad case is s² − t²u equal to zero, which happens when the denominator is s ± √(s²), that is when the original denominator was already zero or the radical was a perfect square that exactly cancelled the whole term. The expression is undefined and the tool says so rather than dividing by zero. It is worth checking for this before you start any conjugate multiplication by hand, because the working looks perfectly normal right up until the last line.
Cube Roots, Mixed Surds, and the Limits of This Page
The conjugate trick relies on the difference of two squares, so it works for square roots and nothing else. A denominator containing a cube root needs a different multiplier built from the difference or sum of two cubes: to clear ∛2 from a denominator you multiply by ∛4, since ∛2 × ∛4 = ∛8 = 2. For a binomial such as 1 + ∛2 the multiplier is 1 − ∛2 + ∛4, which follows from the factorisation of a sum of cubes. This calculator does not handle those cases, and it says so rather than producing something that looks plausible.
It also declines the case where the numerator and denominator contain genuinely different surds, such as (1 + √2) ÷ (3 + √5). Multiplying by the conjugate there produces a numerator with √2, √5 and √10 in it, which is a legitimate expression but not a simplification, and reporting it as one would be misleading. The requirement that the two radicands reduce to the same surd is stated in the field hint for exactly that reason. Paul's Online Notes on Rational Expressions covers the wider family of simplifications these belong to.
Arb Digital writes and builds pages that show the reasoning behind the answer, which is what keeps readers on a page and brings them back.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Multiplying only the denominator — that changes the value. Both the top and the bottom must be multiplied by the same expression, which is what makes the operation multiplication by 1.
- Squaring the denominator instead of using the conjugate — (s + t√u)² still contains a surd in its cross term, so it does not clear anything.
- Getting the conjugate's sign wrong — the conjugate of 3 − √5 is 3 + √5. Only the sign between the terms flips, never the sign of s.
- Forgetting to simplify the radical first — leaving √50 as it is makes every subsequent number five times larger than it needed to be.
- Not reducing the final fraction — (15 − 5√5) ÷ 20 should be written as (3 − √5) ÷ 4. The common factor must divide all three parts, including the denominator.
Related Free Tools From Arb Digital
Simplify a single radical with the square root calculator, reduce the result to lowest terms with the simplify fractions calculator, combine fractions using the fraction calculator, or turn a decimal answer back into exact form with the decimal to fraction calculator. The full free online tools hub lists every algebra calculator we publish.
Frequently Asked Questions
It means rewriting a fraction so that no irrational number appears underneath, usually by multiplying the top and the bottom by the same carefully chosen expression. The value does not change, only the way it is written.
The conjugate of s plus t times the square root of u is s minus t times the same square root. Multiplying the two gives a difference of squares, which contains no radical.
The value is the same either way. Rationalized form is expected as standard, makes expressions easier to compare and combine, and often reveals cancellation that the original form hides.
No. You are multiplying by an expression divided by itself, which equals one. The decimal value is identical before and after, which is why this page prints both as a check.
Multiply by whatever brings the radicand up to a perfect cube. To clear the cube root of 2 you multiply by the cube root of 4, because their product is the cube root of 8, which is 2.
Multiply the numerator and denominator by minus one so the denominator is positive. That flips the sign of every term on top and is purely cosmetic.
Yes, and it is a standard step in calculus when finding limits involving a difference of square roots. The same conjugate method is used, applied to the top rather than the bottom.
This page explains an algebraic method for educational purposes only, and results should be checked against your own working before being used in assessed coursework.
