How this instrument works
Rationalizing a denominator means rewriting a⁄√b so that no root sign survives on the bottom of the fraction. The trick is to multiply the whole fraction by √b⁄√b — a disguised form of 1, since any number over itself equals one — which leaves the fraction's value unchanged while turning the denominator into √b × √b. That product equals b exactly, because a square root is defined as precisely the number that squares back to its original value, so the radical cancels itself out of the bottom and reappears, tamed, in the numerator as a√b.
The habit is older than calculators. Long division by a whole number is mechanical; long division by an endless, non-repeating decimal like √3 = 1.7320508… is not. Clearing the denominator moved every hard division onto a rational divisor, which is why 5⁄√3 was traditionally rewritten as 5√3⁄3 before anyone reached for the actual decimal value. Modern algebra courses kept the convention as the definition of a fully simplified answer, even though a calculator now handles either form with equal ease.
The technique here covers the single-radical case only — a denominator that is one square root, nothing more. A denominator with two terms, such as √2 + 1, does not clear under multiplication by its own radical; it needs its conjugate instead, using the difference-of-squares identity so the middle terms cancel. And if b happens to already be a perfect square, the 'irrational' denominator was never irrational at all — the arithmetic still runs correctly, it just simplifies further afterward.
- Enter the whole-number top of your fraction into Numerator — this is the a in a⁄√b.
- Enter the value sitting under the square-root sign into "Number under the radical, b" — this is the b in a⁄√b.
- Read Rationalized numerator (a√b) for the new top of the fraction, now carrying the radical instead of the bottom.
- Read Rationalized denominator for the new bottom — always the whole number b, with no root sign left in it.
- Write the two results together as one fraction, a√b⁄b, for the final simplified answer.
Worked example — rationalizing 5⁄√3
Start with the fraction 5⁄√3, so the numerator is a = 5 and the number under the radical is b = 3. Multiplying top and bottom by √3 clears the root from the bottom: the denominator becomes √3 × √3 = 3, a whole number, while the numerator becomes 5 × √3 = 8.660254037844386.
The rationalized denominator, 3, matches the field exactly, and the rationalized numerator, 8.660254037844386, is what the sheet returns for a√b. Written as a fraction, that is 8.660254037844386⁄3 — the same value as 5⁄√3, about 2.886751 either way — but now dividing by the whole number 3 instead of by an endless decimal, which is the form an algebra class marks as fully simplified.
Questions
Why bother clearing the square root off the bottom of a fraction?
Because a whole-number denominator is far easier to compute with by hand and is the form algebra courses treat as fully simplified. Dividing 8.660254037844386 by 3 is direct long division; dividing 5 by the endless, non-repeating decimal √3 is not. The convention predates pocket calculators and has simply stuck as the expected final answer.
What is actually happening when you multiply by √b?
Multiplying the fraction by √b⁄√b changes nothing about its value, since that fraction equals 1, but it turns the denominator into √b × √b, which equals b exactly because a square root is defined as the number that squares back to b. The radical does not disappear — it moves to the numerator, where an irrational value is easy to leave in place.
Does this same trick clear a denominator with two terms, like √2 + 1?
No. A two-term, or binomial, denominator needs its conjugate rather than its own radical, because (√2 + 1)(√2 − 1) = 2 − 1 = 1 uses the difference-of-squares identity to cancel the root, while multiplying by √2 alone does not. This calculator handles only the single-radical case a⁄√b; a binomial denominator is a related but separate problem.
What if b is already a perfect square, like 4 or 9?
Then the fraction was never truly irrational — √4 equals 2 exactly. The rationalizing arithmetic still runs correctly, though: with a = 5 and b = 4, this calculator reports a numerator of 10 and a denominator of 4, which then reduces further, by canceling a common factor of 2, to the plain fraction 5⁄2.
What is the most common mistake when doing this by hand?
Multiplying only the denominator by √b and leaving the numerator untouched, which changes the fraction's value instead of preserving it. Both top and bottom must be multiplied by the same √b, because that is identical to multiplying the whole fraction by √b⁄√b — a stand-in for 1 that changes the form but never the amount.
Can the number under the radical, b, be negative?
Not for a real answer — a negative b has no real square root, only an imaginary one, since √−3 equals i√3, which sits outside ordinary rationalizing. The numerator a can be positive, negative, or zero, and whatever sign it carries passes straight through unchanged into the rationalized numerator.