If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra 2>Unit 2

Lesson 1: The imaginary unit i

# Simplifying roots of negative numbers

Discover the magic of the imaginary unit 'i'! This lesson dives into simplifying the square root of negative numbers using 'i', the principal square root of -1. We'll explore how to rewrite negative numbers as products, and use prime factorization to simplify roots. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• what's PRINCIPLE square root??
• Wardah,
For real numbers
The Principal Square Root is the positive square root.
√9 is both -3 and +3
But the Principal square root of 9 is only 3

For imaginary numbers,
the Principal square root of √(-1) is i and not -i

I hope that helps.

It is like saying the absolute value of the square root.
• At he said that it's impossible to use this property when both of them are negative!
I do actually not get that point! Is there any prove? I mean it can't just be by definition, and if it is, why is it allowed to say so? I'm confused!
• Here is the proof I posted for someone else who asked the same question a while ago:
Great question.
*Note that ⁺√ implies principal square root.*
We must prove that:
⁺√(ab) ≠ ⁺√a • ⁺√b
For
a, b < 0
If a and b are negative, then the square root of them must be imaginary:
⁺√a = xi
⁺√b = yi
x and y must be positive (and of course real), because we are dealing with the principal square roots.
⁺√a • ⁺√b = xi(yi) = -xy
-xy must be a negative real number because x and y are both positive real numbers.
On the other hand,
⁺√(ab) = √[(xi)²(yi)²] = (xyi²)² = (xy)²
Since ⁺√(ab) = (xy)² and ⁺√a • ⁺√b = -xy, our problem becomes to prove that:
(xy)² ≠ -xy
For
x, y > 0
Well this is easy! The left hand side is obviously positive and the right hand side is obviously negative, so they cannot be equal! Therefore, ⁺√(ab) ≠ ⁺√a • ⁺√b if a, b < 0. Q.E.D. Comment if you have any questions.
• What is the difference between square root and principal square root?
• The principal square root is always positive. For example, both 3^2 and (-3)^2 equal 9, but the principal square root of 9 is only 3.
• Why can't we continue the line of reasoning Sal mentioned in the video. More specifically, why doesn't it work? I followed it along and eventually got to the conclusion that √52 = -√52, which makes no sense. If this were true, -1 would = 1, and all sorts of weird stuff would happen. What is wrong with that line of reasoning?
• Hello,
√52 can be written as
1) √ (+13).(+4) and
2) √(-13).(-4)
Now expression in (1) follows the property √a.b=√a.√b (or √b.√a)
But in (2) expression fails to be follow this property correctly.
Here we can go further to write
√52=(√-13).(√-4) Now, this is where Sal says it's not right to simplify the square root.
As √-13 can be written as √13.√-1 and similarly √-4 can be written as √4.√-1 and we know i=√-1.
√52=√13.√4.i²
√52=√13.√4(-1) (As i²= -1)
√52=-√52 which isn't right.
• what is the principle square root
• it's the positive square root.
for example, the square roots of 9 would be 3 and -3, but the principal root is the positive one, which is 3
• when we solve a quadratic equation and in case we get the discriminant negative( i know we are gonna get complex solutions) but what do these complex solutions signify on a graph??
• That's a great question! First you need to remember what the solutions normally mean. Usually what you are trying to do is to find the x intercept. That "signifies on the graph" something you probably can understand and look at. But when the solution is complex and you are trying to think of the significance first ask yourself, how do I graph complex numbers? The answer is you need to invent a whole new concept of numbers and this thingy called the complex plane. Read this http://www.purplemath.com/modules/complex3.htm article which puts it a little differently and even has some pictures, that will solidify the concepts a little better in your brain. Complex numbers are very complex and take most people a lot of effort to understand, don't give up.
• if i*√4*√13 is 2i*√13, then why does it is not also -2i*√13?
• We always do the principal root (the positive root) unless there is already a "-" in front of the radical.
√4 = 2 because it is asking for the positive root
-√4 = -2 because it is asking for the negative root

Hope this helps.
• Are complex numbers same as imaginary numbers ?
• Complex numbers: all numbers of the form a + bi with real-values a and b where i is the imaginary unit.

Imaginary numbers: all numbers of the form bi with real-valued nonzero b. A subset of the complex numbers.
• √-77 doesn't have any perfect squares in it what should I do?
• You do the square root of -1, and leave the 77 inside the radical.
√(-77) = √(77)i