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### Course: Algebra 2 > Unit 2

Lesson 1: The imaginary unit i- Intro to the imaginary numbers
- Intro to the imaginary numbers
- Simplifying roots of negative numbers
- Simplify roots of negative numbers
- Powers of the imaginary unit
- Powers of the imaginary unit
- Powers of the imaginary unit
- i as the principal root of -1

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# 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??(47 votes)
- 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.(104 votes)

- At2:00he 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!(45 votes)- 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.(53 votes)

- What is the difference between square root and principal square root?(11 votes)
- 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.(15 votes)

- 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?(16 votes)
- 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.

So,√52=(√13.i).(√4.i), which leads to

√52=√13.√4.i²

√52=√13.√4(-1) (As i²= -1)

√52=-√52 which isn't right.(5 votes)

- what is the principle square root(8 votes)
- 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(16 votes)

- 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??(6 votes)
- 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.(9 votes)

- Are complex numbers same as imaginary numbers ?(4 votes)
- 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.(9 votes)

- if i*√4*√13 is 2i*√13, then why does it is not also -2i*√13?(4 votes)
- 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.(9 votes)

- √-77 doesn't have any perfect squares in it what should I do?(4 votes)
- You do the square root of -1, and leave the 77 inside the radical.

√(-77) = √(77)i(8 votes)

- What does he mean when he keeps saying the "principal square root"? He corrected himself multiple times in the video so he was saying that not just "square root".(2 votes)
- Given the number 4 for example, the square root of 4 could be 2 or it could be -2.

The principal square root is the positive one.

A great website to get to know, Wolfram Alpha, has this as the definition of the principal square root.

http://mathworld.wolfram.com/PrincipalSquareRoot.html(6 votes)

## Video transcript

We're asked to simplify
the principal square root of negative 52. And we're going to assume,
because we have a negative 52 here inside of the
radical, that this is the principal branch of the
complex square root function. That we can actually put,
input, negative numbers in the domain of this function. That we can actually get
imaginary, or complex, results. So we can rewrite negative
52 as negative 1 times 52. So this can be rewritten as
the principal square root of negative 1 times 52. And then, if we assume that
this is the principal branch of the complex square root
function, we can rewrite this. This is going to be equal to
the square root of negative 1 times-- or I should say,
the principal square root of negative 1 times the
principal square root of 52. Now, I want to be
very, very clear here. You can do what we just did. If we have the principal square
root of the product of two things, we can rewrite that
as the principal square root of each, and then
we take the product. But you can only do
this, or I should say, you can only do this if
either both of these numbers are positive, or only
one of them is negative. You cannot do this if both
of these were negative. For example, you
could not do this. You could not say the
principal square root of 52 is equal to negative
1 times negative 52. You could do this. So far, I haven't
said anything wrong. 52 is definitely negative
1 times negative 52. But then, since these
are both negative, you cannot then say that this
is equal to the square root of negative 1 times the
square root of negative 52. In fact, I invite
you to continue on this train of reasoning. You're going to get
a nonsensical answer. This is not OK. You cannot do this,
right over here. And the reason why you cannot do
this is that this property does not work when both of
these numbers are negative. Now with that said, we
can do it if only one of them are negative or both of
them are positive, obviously. Now, the principal square
root of negative 1, if we're talking about
the principal branch of the complex square
root function, is i. So this right over here
does simplify to i. And then let's think
if we can simplify the square root of 52 any. And to do that, we can think
about its prime factorization, see if we have any perfect
squares sitting in there. So 52 is 2 times 26,
and 26 is 2 times 13. So we have 2 times
2 there, or 4 there, which is a perfect square. So we can rewrite this as equal
to-- Well, we have our i, now. The principal square
root of negative 1 is i. The other square root of
negative 1 is negative i. But the principal square
root of negative 1 is i. And then we're going
to multiply that times the square
root of 4 times 13. And this is going to be equal
to i times the square root of 4. i times the square root of 4,
or the principal square root of 4 times the principal
square root of 13. The principal square
root of 4 is 2. So this all simplifies, and
we can switch the order, over here. This is equal to 2 times
the square root of 13. 2 times the principal square
root of 13, I should say, times i. And I just switched
around the order. It makes it a little
bit easier to read if I put the i after
the numbers over here. But I'm just multiplying i times
2 times the square root of 13. That's the same thing
as multiplying 2 times the principal square
root of 13 times i. And I think this is about as
simplified as we can get here.