How this instrument works
Rationalizing a denominator means rewriting a⁄√b so the radical no longer sits underneath the fraction — multiplying both the top and bottom by √b clears it out, since √b times √b equals b exactly, leaving a√b⁄b instead. Nothing about the fraction's actual value changes; only where the radical sits within it does.
This is a different simplification MOTIVE from this site's Radical calculator, which pulls a perfect-square factor out from INSIDE a root (√(a²b)=a√b). Here, the goal is moving a radical that's sitting in the DENOMINATOR up into the numerator instead — traditionally considered the properly 'simplified' place for it to live, since dividing by an irrational number is awkward to do by hand compared to multiplying by one.
Both forms are computed directly here and compared, rather than simplified only algebraically — a quick, concrete way to confirm the rationalizing step really is value-preserving, not just a symbolic manipulation.
- Enter the fraction's numerator into the a field.
- Enter the number under the radical (in the denominator) into the b field.
- Read a ⁄ √b: the original, un-rationalized form.
- Read a√b ⁄ b: the rationalized form, and confirm the two values match.
Worked example — rationalizing 3⁄√5
With a=3 and b=5: the original form, 3⁄√5, comes out to about 1.342. The rationalized form, 3√5⁄5, also comes out to about 1.342 — confirming that multiplying top and bottom by √5 (clearing the radical from the denominator) doesn't change the fraction's actual value, only how it's written.
With a=10 and b=2: 10⁄√2≈7.071, and its rationalized form, 10√2⁄2, also ≈7.071. A case needing no real work at all: a=1, b=16 gives 1⁄√16=1⁄4=0.25 directly, and its rationalized form, 1×4⁄16, also comes out to 0.25 exactly — since 16 is already a perfect square, there was no true radical left in the denominator to clear in the first place.
Questions
What does it mean to rationalize a denominator?
Rewriting a fraction so no radical sign remains in the denominator, by multiplying both the top and bottom by that same radical — since a radical times itself gives a plain rational number, the denominator becomes radical-free.
Why bother rationalizing a denominator at all?
It's traditionally considered the properly simplified form, and it's genuinely easier to work with by hand: dividing by an irrational number is awkward, while multiplying by one (once it's moved to the numerator) is straightforward.
Does rationalizing change the fraction's value?
No — multiplying both the numerator and denominator by the identical nonzero quantity never changes what the fraction is worth, only how it's written, which this page confirms directly by computing both forms and comparing them.
How is this different from the Radical calculator on this site?
That page pulls a perfect-square factor out from INSIDE a radical (√(a²b)=a√b); this page instead clears a radical that's sitting in a fraction's DENOMINATOR, a related but distinct simplification goal.
What if the denominator is already a perfect square?
Then there's no true radical left to clear in the first place — the square root simplifies to a whole number on its own, and rationalizing changes nothing beyond confirming that already-simple result.