How this instrument works
Simplifying a radical means spotting a perfect-square piece hiding inside it and moving that piece's own square root outside the radical sign, leaving a smaller number behind underneath. The identity behind that move is √(a²b) = a√b, valid whenever a is zero or positive, and it is really just the ordinary rule for splitting a product under one radical into two radicals, √(xy) = √x × √y, applied to the specific case where x is a perfect square, a². Once √(a²) simplifies to plain a, only √b is left needing a radical sign at all.
This calculator checks the identity rather than merely applying it once and moving on — it reports both routes side by side as left and right, so agreement between them is something you see rather than something you take on faith. That sets it apart from a page on this site that multiplies two separate, independent radicals together into one combined result; here there is only ever one radical to begin with, and the question is whether pulling part of it outside changes anything. It is also distinct from a bare square-root sheet that returns a single value for one number — this one exists specifically to compare two computation paths for a single shared answer.
Set a to 1 and there is no perfect-square piece worth extracting at all — a² is just 1, so left and right both reduce to the same plain √b, and the identity holds without doing any visible simplifying work. That trivial case matters precisely because it shows the identity is not a trick that only works when a simplification happens to exist; it is a statement that remains true even when nothing under the root actually needed pulling out.
The identity's proof is short: √(a²b) can be split into √(a²) × √b, and since a is zero or positive, √(a²) reduces to exactly a, leaving a × √b. Every pair of values run through this sheet is really the same three-step argument confirmed with real numbers standing in for a and b, which is why left and right always agree to as many digits as the calculation carries.
- Enter the perfect square's own root into a — the number being pulled outside the radical.
- Enter whatever remains under the radical into b — the factor that stays put after a comes out.
- Read left, √(a²b), for the value computed directly, without any simplifying step taken first.
- Read right, a√b, for the value computed by the already-simplified route.
- Compare left and right — matching to the same decimal digits is the identity itself, confirmed rather than assumed.
Worked example — simplifying √45 down to 3√5
Set a=3 and b=5. Left computes √(3²×5) = √(9×5) = √45, which comes out to approximately 6.708203932499369. Right computes 3×√5 instead: √5 is about 2.2360679774997896, and 3 times that is 6.708203932499369 as well — the same eleven-plus digits, reached by an entirely different arithmetic path. Nine, the perfect square hiding inside 45, is exactly what let its own root, 3, step outside the radical while 5 stayed behind underneath.
Two further pairs confirm the pattern holds at its edges and beyond. With a=1 and b=7, nothing is being extracted at all, and left and right both settle on √7, roughly 2.6457513110645907. With a=4 and b=2, left computes √(16×2) = √32, and right computes 4×√2 — both approximately 5.656854249492381, matching even though the extracted factor, 16, is noticeably larger than in the first pair.
Questions
What does it mean to simplify a radical like this?
It means recognizing that a perfect square sits inside the number under the root, then moving that perfect square's own root outside the radical sign while a smaller factor stays underneath. √45 becomes 3√5 because 45 contains 9, a perfect square whose root is 3, leaving 5 behind under the remaining radical.
Why does √(a²b) equal a√b?
Because a product under one radical can always be split into two radicals multiplied together, √(a²b) = √(a²) × √b, and √(a²) reduces to plain a whenever a is zero or positive. Multiplying that a back by √b gives a√b, the same value the unsplit expression started with.
What happens when a is set to 1?
Nothing is actually pulled out, since 1² is just 1. Both left and right collapse to the same plain √b — with b=7, both sides read approximately 2.6457513110645907 — showing the identity holds even in the case where no simplification takes place.
Is this the same operation as multiplying two radicals together?
No. Multiplying two radicals combines two separate, independent square roots into one, √a × √b = √(ab). This sheet starts from a single radical, √(a²b), and asks whether part of it can be moved outside without changing the value — a different question working in the opposite direction from combining two roots.
Does the identity still hold if a is negative?
Not in the form shown here, since √(a²) equals the non-negative value of a, not a itself, once a drops below zero. This sheet keeps a at zero or above so that √(a²) and a stay identical, which is exactly what lets the identity read as a clean equality rather than needing an extra absolute-value step.
How can I spot a perfect-square factor worth pulling out of a radical?
Check whether the number under the root divides evenly by a perfect square such as 4, 9, 16, or 25. Forty-five divides evenly by 9, so √45 splits into √9 × √5, and √9 simplifies to 3, leaving 3√5 — the same result this sheet returns as right when a=3 and b=5.