When √a × √b is not √(ab).

√a and √b drawn as points on the Argand plane (real for a ≥ 0, on the imaginary axis for a < 0); their product is compared with √(ab) and the mismatch lights up when both are negative.

Find a pair with √a × √b ≠ √(ab) (both negative, e.g. a = −4, b = −9 gives −6 vs 6).

Writing √−4 × √−9 = √36 = 6.

• Both i and −i square to −1, since (−i)² = i² = −1. So −1 has two square roots, i and −i, and both solve x² = −1. The symbol √−1 is kept for i alone.
• In the same way (√3 i)² = 3i² = −3 and (−√3 i)² = −3, so the square roots of −3 are √3 i and −√3 i, and the symbol √−3 means √3 i.
• For any positive real a, √−a = √a × √−1 = √a i. For example √−16 = 4i and √−7 = √7 i.
• The rule √a × √b = √(ab) holds when a and b are both positive, and also when one is positive and the other negative. If either is zero, both sides are 0.
• The rule fails when a and b are both negative: √−4 × √−9 = 2i × 3i = 6i² = −6, but √((−4)(−9)) = √36 = 6. Applying the rule to √−1 × √−1 would give √1 = 1, contradicting i² = −1.
• Safe habit: write each root of a negative number as √a i first, and only then multiply.

Teaching this idea? Put the simulation on your own page or LMS with this code.