Free Multiplying Radicals Calculator

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Multiplying two square roots is one of the fundamental radical rules in algebra: √a × √b = √(ab), meaning two separate roots can be combined into a single radical covering the product of both numbers underneath. This rule is a staple of algebra and geometry coursework — it's exactly what's used when simplifying expressions involving the Pythagorean theorem, distance formulas, or any calculation that produces multiple square roots that need to be combined into one simplified answer.

How it works

Enter the two radicands (the numbers under each square root). The calculator multiplies them together to get a single number under one radical, √(a×b), then searches for the largest perfect square factor of that product — dividing it out and moving its square root outside the radical to produce the fully simplified form.

Enter first radicand (a) under √.
Enter second radicand (b) under √.
Click Calculate to see your results.

Examples

√8 × √2

√8 × √2 = √(8×2) = √16 = 4 — the product happens to be a perfect square, so it simplifies all the way to a whole number.

√6 × √10

√6 × √10 = √(6×10) = √60. The largest perfect square factor of 60 is 4 (60=4×15), so √60 = √4×√15 = 2√15.

Common mistakes to avoid

Multiplying the radicands but forgetting to check for a perfect square factor in the simplification step, leaving an unsimplified radical as the final answer.
Adding the radicands instead of multiplying them (√a × √b ≠ √(a+b)).
Attempting the operation with a negative radicand, which doesn't have a real square root.

Frequently asked questions

No — both radicands must be non-negative for the product to have a real (non-imaginary) square root result. Taking the square root of a negative number requires venturing into complex numbers.

It follows directly from the definition of square roots: (√a × √b)² = (√a)² × (√b)² = a × b, so √a × √b must equal the (non-negative) square root of a×b, which is √(ab).

Check the product against perfect squares in decreasing order (4, 9, 16, 25, ...) to see which is the largest one that divides it evenly — for 60, that's 4 (since 60=4×15 and 15 has no larger perfect square factor).

That's completely normal and even common — √8 and √2 are both irrational on their own, but their product √16=4 is a whole number, since irrational factors can combine to cancel each other out.

Yes — √a × √b × √c combines into a single √(abc), and the same perfect-square-factor simplification process applies to the resulting product.

Since √1=1, multiplying by it doesn't change the other radical at all — the result is just the other square root, simplified as usual.

No — multiplication is commutative, so √a × √b always gives the identical result as √b × √a.