Main content

## The imaginary unit i

# Simplifying roots of negative numbers

CCSS.Math:

## 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.