(square root of -4) * ( square root of -9) = -6? What on Earth?

Level 2

On this page -- that I saw advertised on reddit -- I was asked a question that I used the Python programming language to get my computer to figure out. Unless I'm missing something, the answer (of 6) that I gave was correct. Please could someone explain to me what I'm missing?

If I'm not missing anything; what? Am I supposed to take their word for it over my computer's? If anyone wants to take it up -- if I'm not missing anything -- you can argue with the code. The computer doesn't lie:

answer = (-4 ** (1 / 2)) * (-9 ** (1 / 2))
print(answer)
print(answer == -6)

Thank you for the help. :)

-6i -6 6i 6

1 solution

Brian Charlesworth
Dec 9, 2017

$\sqrt{-4} \times \sqrt{-9} = \sqrt{-1 \times 4} \times \sqrt{-1 \times 9} = 2i \times 3i = 6 \times i^{2} = 6 \times -1 = \boxed{-6}$ .

Perhaps your calculation went $\sqrt{-4} \times \sqrt{-9} = \sqrt{-4 \times -9} = \sqrt{36} = 6$ . This is not valid, however, since $(ab)^{c} = a^{c}b^{c}$ only works (all the time) for $a,b \ge 0$ . Part of the motivation for this restriction is that we want a unique value for something like $\sqrt{36}$ , and without this restriction we would have $\sqrt{36}$ equalling both $6$ and $-6$ , which is problematic.

@Brian Charlesworth

Thank you for the help. :)

Arthur Heuer - 3 years, 5 months ago

(square root of -4) * ( square root of -9) = -6? What on Earth?

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