This is slightly tricky, since sqrt(a)sqrt(b) = sqrt(ab)√a√b=√ab is only generally true for a, b >= 0a,b≥0.
If you thought it held for negative numbers too then you would have spurious 'proofs' like:
1 = sqrt(1) = sqrt(-1*-1) = sqrt(-1)sqrt(-1) = -11=√1=√−1⋅−1=√−1√−1=−1
Instead, use the definition of the principal square root of a negative number:
sqrt(-n) = i sqrt(n)√−n=i√n for n >= 0n≥0, where ii is 'the' square root of -1−1.
I feel slightly uncomfortable even as I write that: There are two square roots of -1−1. If you call one of them ii then the other is -i−i. They are not distinguishable as positive or negative. When we introduce Complex numbers, we basically pick one and call it ii.
Anyway - back to our problem:
sqrt(-50) * sqrt(-10) = i sqrt(50) * i sqrt(10) = i^2 * sqrt(50)sqrt(10)√−50⋅√−10=i√50⋅i√10=i2⋅√50√10
= -1 * sqrt(50 * 10) = -sqrt(10^2 * 5) = -sqrt(10^2)sqrt(5)=−1⋅√50⋅10=−√102⋅5=−√102√5
= -10sqrt(5)=−10√5
