Question 1133226: what is the correct solution?
√-80 (square root of -80)
a.i√-80
b.-4i√5
c.-80i
d.4i√5
*√ (symbol of a square root)
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
Answer d.
Note that both answer b and answer d, when squared, give the same result of -80. However, just as 2 and -2 squared both give the result 4 but sqrt(4) = 2 and not -2, the square root of -80 is 4i*sqrt(5) and not -4i*sqrt(5).
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Added after reading the response by tutor @ikleyn....
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It is true that, in the complex domain, every number, real or complex, has two square roots. So the square root of 4 has two square roots, 2 and -2; and the square root of -80 has two square roots, 4i*sqrt(5) and -4i*sqrt(5).
But when you write the number sqrt(4), or sqrt(-80), for use in a calculation, it has to have a single value.
A high school student who wrote as an answer on a test that the square root of 4 is -2 would be marked wrong; and if he used -2 as the value of the square root of 4 in a calculation, he would end up with the wrong answer.
In this problem, two of the answer choices are 4i*sqrt(5) and -4i*sqrt(5). Since it would be presumed that these are answer choices on a test with only one correct answer, the answer should be d only.
Answer by ikleyn(52772)  (Show Source):
You can put this solution on YOUR website! .
In complex domain, square root of ANY non-zero complex number has 2 (two, TWO) values.
The correct answers are b) and d) :  and  .
The square of any of these two complex numbers is equal to -80.
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Added after reading the post by @greenestamps.
 also has two values: 2 and -2.
When in the school arithmetic you use only "positive" values of square roots, like
2*sqrt(7) + 9*sqrt(7) = 11*sqrt(7),
it is the result of an agreement (often accepted as an implicit agreement or a context) specially for the given calculation.
But in general, the square root of any positive real number has two values, one positive and the other negative.
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The truth is that the square root from a number IS NOT a single number: it is a set of two numbers.
In the school Math, the teachers hide this truth from students.
They do it due to two reasons:
a) They themselves can not explain it to students.
b) The average school student will not understand it.
But the truth does not stop to be truth of it.
In the university, in advanced courses of algebra and complex analysis, the professors will explain it to you.
And then this truth will become absolutely clear to you : as 2 x 2 = 4. Or as a fact that the Earth is spherical - not flat . . .
Then recall my post . . .
In school, the teacher can explain only half of the truth to you . . .
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