Can someone explain how the square root of two negative numbers multiplied together doesn't equal a positive/negative number.

Question

On a test I got a question asking to simplify $\sqrt{-25}\times \sqrt{-3}$

I answered $5\sqrt 3$ but apparently it's only $-5\sqrt 3$ because of some rule of imaginary numbers, could someone please explain to me why $\sqrt{-25}\times \sqrt{-3} \ne \pm 5\sqrt 3$

One issue is to choose consistently the sign of the square root ("branch cut" and "Riemann Surface" are two different approaches). But even these approaches don't guarantee to preserve intuitive arithmetic. If you want to take square (or other) roots of negative (and also complex) numbers you simply have to take care and be aware that it can't be done in a way which preserves every property you first thought of. — Mark Bennet, Feb 12 '21 at 19:20
@MarkBennet Indeed, but this is almost certainly beyond the scope of OP's learning at the moment :) — diracdeltafunk, Feb 12 '21 at 19:22
@diracdeltafunk I hit the return key at the wrong time - I did want my comment to indicate the richness of the challenge and not to be mystifying, I hope my edit has improved it somewhat. — Mark Bennet, Feb 12 '21 at 19:24
See Why doesn't multiplying square roots of imaginary numbers follow $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$? — , Feb 12 '21 at 19:27
This is a great question, but it has been asked many times on this site. See e.g. here. — Noah Schweber, Feb 12 '21 at 19:27

score 3 · Answer 1 · answered Feb 12 '21 at 19:21

3

You presumably tried to use the "rule" $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. However, this only holds when $a$ and $b$ are positive! It is not applicable here.

Instead, we need to compute this product directly. $\sqrt{-25} = 5i$ and $\sqrt{-3} = i\sqrt{3}$, so $$\sqrt{-25} \sqrt{-3} = (5i)(i \sqrt{3}) = -5\sqrt{3}.$$

answered Feb 12 '21 at 19:21

diracdeltafunk

14,911

But −5√3̅ = -5√̅1̅̅3̅ and the √1̅ = ±1 if we use the √1̅ = -1 and we can then say that 5√̅1̅̅3̅ = -5√̅(̅-̅1̅)̅(̅-̅1̅)̅*̅3̅ then we can either take out the -1's to get -(-1)5√̅3̅ or we can turn the -1's to i² and take them out to get i²5√̅3̅ both of which are equal – Dalton Feb 12 '21 at 19:41
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@Dalton $\sqrt{1}$ denotes the positive square root of $1$, not both of them. The square roots of $1$ are $\pm \sqrt{1}$. It's not the case that $\sqrt{1}=\pm 1$. – MHW Feb 12 '21 at 20:03

Can someone explain how the square root of two negative numbers multiplied together doesn't equal a positive/negative number.

1 Answers1