How this instrument works
√a × √b and √(ab) are the same number whenever a and b are zero or positive, and the reason sits in one property of squaring: square the product √a × √b and the factors regroup as (√a × √a) × (√b × √b) — multiplication doesn't care what order you multiply in — which works out to a × b. A nonnegative number whose square is a × b can only be √(ab) itself, since a nonnegative real number has exactly one nonnegative square root. The two expressions have to match, term for term, without either root ever being evaluated first.
That collapsing is more than tidy bookkeeping — it regularly turns two individually irrational numbers into an exact integer. √3 is roughly 1.7320508 and √12 is roughly 3.4641016, and multiplying those two decimals by hand looks like it should hand back more decimals. Instead, 3 × 12 = 36 is a perfect square, so √3 × √12 = √36 = 6 on the nose. Two numbers with no terminating decimal representation multiply to a whole number, and the identity is the reason it isn't a coincidence.
The rule needs both radicands nonnegative to stay inside the real numbers. Set a to 0 and the whole product falls to 0 regardless of b, since √0 is 0 and anything times 0 is 0. Push a radicand negative, though, and the shortcut breaks in a way that trips up algebra students specifically: √-1 × √-1 is not √1 = 1; each √-1 is defined as the imaginary unit i, so the true product is i × i = -1. √a × √b = √(ab) is a real-number statement — it stops applying the moment a or b crosses zero.
- Enter the number under the first radical into a, under first radical — this is a in √a × √b, defaulting to 3.
- Enter the number under the second radical into b, under second radical — any value zero or greater works, defaulting to 12.
- Read √a × √b for the already-combined, already-simplified product — no separate square-root lookups are needed.
- Try pairs where a × b is a perfect square, like 3 and 12 or 2 and 8, to watch two irrational-looking roots collapse to a whole number.
Worked example — √3 times √12
Set a to 3 and b to 12. Taken separately, √3 ≈ 1.7320508 and √12 ≈ 3.4641016 — two decimals that don't look related to anything clean. The identity says multiply what's under the radicals first: 3 × 12 = 36, and 36 is a perfect square, so √3 × √12 = √36 = 6 exactly. The field labelled √a × √b reads 6.
Multiplying the two decimals directly agrees to the limit of the display: 1.7320508 × 3.4641016 = 6.0000000, with any rounding noise living only in how many digits were carried, not in the answer itself. Combine-first and multiply-first are the same real number reached by different arithmetic — one route just skips the decimals entirely.
Questions
What does the identity √a × √b = √(ab) actually say?
It says two square roots multiplied together equal the square root of their radicands multiplied together — the product of the roots is the root of the product. It holds whenever a and b are both zero or positive, which keeps every root in the real numbers rather than the imaginary ones.
Why is it valid to merge two separate square roots into one radical?
Square both sides: (√a × √b)² regroups to (√a × √a) × (√b × √b), which is a × b. So √a × √b is a nonnegative number whose square is a × b, and √(ab) is the only nonnegative number with that property — there's no other candidate it could be.
Will multiplying two irrational-looking radicals ever produce a whole number?
Yes, whenever the product of the two radicands is itself a perfect square. √3 × √12 = √36 = 6 exactly, and √2 × √8 = √16 = 4 exactly — in both cases the individual roots are irrational, but their product isn't, because 3 × 12 and 2 × 8 both happen to land on perfect squares.
What happens if a or b is zero, or if either one is negative?
If either radicand is 0, the product is always 0, since √0 = 0. Negative radicands leave the real numbers entirely: √-1 is the imaginary unit i, not a real number, so √-1 × √-1 = i × i = -1 rather than √(-1 × -1) = √1 = 1 — the identity only governs nonnegative a and b.
Does the same combine-into-one-radical trick work for addition, √a + √b?
No, and mixing the two up is the most common mistake with radicals. Multiplication and division pass through a square root cleanly, but addition does not: check a = 4, b = 9, where √4 + √9 = 2 + 3 = 5, while √(4 + 9) = √13 ≈ 3.606 — two different numbers.
Does the rule still apply if the two radicals have different indices?
Not directly. √a × √b = √(ab) is a same-index rule for two square roots. Multiplying a square root by a cube root — a^(1/2) times b^(1/3) — needs a common index first, usually by rewriting both as sixth roots, before the radicands can be combined under one symbol.