Main content

## Algebra (all content)

### Unit 11: Lesson 8

Radicals (miscellaneous videos)- Simplifying square-root expressions: no variables
- Simplifying square roots of fractions
- Simplifying rational exponent expressions: mixed exponents and radicals
- Simplifying square-root expressions: no variables (advanced)
- Intro to rationalizing the denominator
- Worked example: rationalizing the denominator
- Simplifying radical expressions (addition)
- Simplifying radical expressions (subtraction)
- Simplifying radical expressions: two variables
- Simplifying radical expressions: three variables
- Simplifying hairy expression with fractional exponents

ยฉ 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Simplifying radical expressions: two variables

A worked example of simplifying elaborate expressions that contain radicals with two variables. In this example, we simplify √(60x²y)/√(48x). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- "why did you convert 5/4 to 1/4*5?" Adding to this question above what videos if any cover the reasons for doing this..I wish to understand it better.

Thanks(5 votes)- Because it makes it easier to simplify, because some people get confused with fractions.(4 votes)

- Isn't the xy also divided by 4, like the five is?

Thanks(5 votes)- They are all the same term. If they weren't, and the numerator was something like 5+xy you wouldn't be able to, or at least not as cleanly. If that was the case it would be 5/4+xy/4 as you indicated... but since it is all one term: 5xy, its just fine. Heres the reasoning:

5xy/4 is the same as 5xy divided by 4. Division by 4 is the same as multiplying by 1/4. When you do 5xy*1/4 you get 5/4xy. Incase that sounds a bit strange, just think what happens when you multiple 2xy by 4. You would get 8xy, works the same way for fractions!(4 votes)

- I have 1875x^9-48x^5, and i need to factor it. How would I do that? I am soo confused!(3 votes)
- First factor out the GCD:

3x^5 ( 625x^4 - 16)

The binomial is a difference of squares:

3x^5 (25x^2+4)(25x^2 - 4)

The second binomial is also a difference of squares:

3x^5 (25x^2+4)(5x+2)(5x-2)

Final Answer.(1 vote)

- Wouldn't x be on the outside of the radical sign because x squared is just x? I'm confused. Can someone please explain how the x's cancel kurt/divide. I re-did the question and that was the only part I had trouble with.(1 vote)
- Sal divided the x^2 in the numerator by the x in the denominator, which leaves an x in the numerator because x^2/x is x. So x^2 isn't there anymore.(3 votes)

- is sqrt(a^2 + b^2) the same as sqrt(a^2) + sqrt(b^2)?(1 vote)
- โ(aยฒ+bยฒ) = โaยฒ + โbยฒ

Let's see if this works when a=3 and b= 4

So,

โ(3ยฒ + 4ยฒ) must equal โ3ยฒ + โ4ยฒ

โ9+16 must equal 3+4

โ25 must equal 7

But โ25 = 5

And 5โ 7

So โ(aยฒ+bยฒ) does not equal โaยฒ+โbยฒ(2 votes)

- Can't you make 5/4 into 1.25, so you just take out the fraction and turn it into a decimal(1 vote)
- You could, but it makes it harder. It is easier to simplify sqrt(5/4) = sqrt(5)/2.(2 votes)

- whats the difference between principal root and square root?(1 vote)
- Principal root is the default and is the non-negative root.

sqrt(9) = 3, not -3

If the problem wants you to use the negative root, then there will be a minus in front of the radical.

- sqrt(9) = -3

Hope this helps.

FYI - Search for intro to square roots. I believe Sal defines the principal root in that vidoe.(2 votes)

- How would you simplify 3x (sqrt(2y))(1 vote)
- Did anyone try to solve this a different way and get an X on the numerator and a radical X on the denominator?(1 vote)
- Would square root 5xy/2 be an equivalent expression?(1 vote)
- Yes, if you multiply 1/2 * sqrt(5xy), you get sqrt(5xy)/2(1 vote)

## Video transcript

We're asked to
divide and simplify. And we have one
radical expression over another radical expression. The key to simplify
this is to realize if I have the principal root of
x over the principal root of y, this is the same thing as the
principal root of x over y. And it really just comes out
of the exponent properties. If I have two things that
I take to some power-- and taking the principal root
is the same thing as taking it to the 1/2 power-- if
I'm raising each of them to some power and
then dividing, that's the same thing as
dividing first and then raising them to that power. So let's apply that over here. This expression
over here is going to be the same thing
as the principal root-- it's hard to write
a radical sign that big-- the principal root
of 60x squared y over 48x. And then we can first look
at the coefficients of each of these expressions and
try to simplify that. Both the numerator and the
denominator is divisible by 12. 60 divided by 12 is 5. 48 divided by 12 is 4. Both the numerator and the
denominator are divisible by x. x squared divided
by x is just x. x divided by x is 1. Anything we divide
the numerator by, we have to divide
the denominator by. And that's all we have left. So if we wanted
to simplify this, this is equal to the--
make a radical sign-- and then we have 5/4. And actually, we can write it
in a slightly different way, but I'll write it
this way-- 5/4. And we have nothing left in the
denominator other than that 4. And in the numerator, we
have an x and we have a y. And now we could leave
it just like that, but we might want to take more
things out of the radical sign. And so one possibility
that you can do is you could say that this is
really the same thing as-- this is equal to 1/4 times 5xy, all
of that under the radical sign. And this is the same
thing as the square root of or the principal
root of 1/4 times the principal root of 5xy. And the square root of
1/4, if you think about it, that's just 1/2 times 1/2. Or another way you
could think about it is that this right here
is the same thing as-- so you could just say,
hey, this is 1/2. 1/2 times 1/2 is 1/4. Or if you don't realize
it's 1/2, you say, hey, this is the same thing
as the square root of 1 over the square root of 4,
and the square root of 1 is 1 and the principal root of 4 is
2, so you get 1/2 once again. And so if you simplify
this right here to 1/2, then the whole thing can
simplify to 1/2 times the principal root--
I'll just write it all in orange-- times the
principal root of 5xy. And there's nothing
else that you can really take out of the
radical sign here. Nothing else here
is a perfect square.