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## Algebra 1

### Unit 11: Lesson 3

Simplifying square roots# Simplifying square roots (variables)

CCSS.Math:

A worked example of simplifying radical with a variable in it. In this example, we simplify 3√(500x³). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Hi all.

Looking for some help with the following:

3/√63

So far I have:

3/√63

3/(√9 x √7)

3/3√7

√7

The answer in the book I'm studying from, however says that the answer is:

√7/7

How does one arrive at this answer?(54 votes)- Hi. When working with radicals, you can't have a radical for a denominator. This is not considered proper in Algebra. So your answer to

3 / √63 would eventually equal 1/ √7.

In order to make it algebraically proper, we have to multiply it by √7 / √7 (which is equal to one, so we know it won't change the value of our equation).

So we get:`1 / √7 * √7 / √7 =`

√7 / √7 * √7

√7 / 7

We know that √ 7 * √7 is the same thing as √ 7*7, or √49, which is 7. That gives us the final answer of √7 / 7 which can't be reduced any farther.

Hope this helps.Sylvia.(108 votes)

- This is hard stuff and he seems to be blowing through it rapidly like it's just review. What is the previous video he keeps referring to in his commentary? The video I see before this one is entitled "Simplifying Square Roots" and he doesn't seem to cover a lot of this. Am I missing something?(53 votes)
- I'm with you. Last vid is the same for me, "Simplifying Square Roots". I watched and understood all of the previous vids leading up to this one and he did blow through this one without slowing down to take the time to really explain why he's doing that. Yet he'll go over extremely simple concepts ad nauseam(52 votes)

- You confused me...Why can't X be a negative number?(37 votes)
**coughs**I'm late 8 years but whatever.

Simple: you can't have negative rads. Use a calculator but a negative number like -1, and it will say "Error" or something like that.

There are no two numbers that when you multiply them twice (like 2 • 2, aka 2^2), you**cant**get a negative number. Like a negative times, a negative is positive, and a positive times a positive is a positive.

And you**cant**multiply two different numbers (negative and positive) in a rad because rads represent a number multiplied by itself to get that number inside the rad.(4 votes)

- I don't understand most of this. I think i'm missing a video.(32 votes)
- I'm sorry, but can we try one WITHOUT an x? I'm going into Grade 11 and never learned this. How do you simplify something like (sq root)45?

In the practice, there are questions where c= sq. root 45 and they need a simplified radical. How do you do that?(19 votes)- I'm gonna use sqrt as square root

sqrt(45) = sqrt(9*5) = sqrt(9) * sqrt(5) = 3 * sqrt(5)

Hope that makes sense! :)(21 votes)

- what happens when the absolute value of a number turns out to be a negative, not a positive(4 votes)
- The absolute value of a number is always positive and is will never be a negative.(34 votes)

- Why does Sal keep saying "the principal root of..." as opposed to "the square root of..."? At1:30I heard him say "the square root of, or the principal root of..." so does that mean they're the same thing? Because it appears as if he sort of corrected himself.(19 votes)
- could you not just do +- (the plus or minus sign -- my computer doesn't let me insert special charactors)?(2 votes)

- If you watch these videos in the order the lessons are published for Algebra basics > foundations > square roots module, there are a lot of references in "Simplifying square roots (variables)" to the previous video (0:25and1:10). But the previous video is about fractions, not variables. I had to jump over to another module (Math > Algebra 1 > Exponents & radicals > Simplifying square roots) to find the video I THINK he's referring to, "Simplifying square root expressions".

Can anyone at Khan Academy explain to me why this learning module would leave out lessons required for completing the content?(14 votes) - I'm at a loss, The quiz is asking questions that the videos have no explanation of. I have x^9, but the video only explains how to factor it one time..(7 votes)
- Matthew,

This video is part of the Algebra I course and is not meant to stand alone. Many of the concepts applied here are explained in previous videos in this course and in the Pre-Algebra course.

Take a look at the outline of Algebra I (https://www.khanacademy.org/math/algebra). You may want to go back to the previous videos in*Rational exponents & radicals*or to the*Exponents, radicals, and scientific notation section*of Pre-Algebra (https://www.khanacademy.org/math/pre-algebra)

The Law of Exponents tells us

x^a * x^b = x^(a+b)

Therefore

x^4 + x^5 = x^(4+5) = x^9

x^4 is a perfect square - x^2 * x^2

You could also consider

x^9 = x * x * x * x * x * x * x * x * x

x^9 = (x * x * x * x) * (x * x * x * x) * x

x^9 = x^4 * x^4 * x(14 votes)

- Around2:18of the video Sal mentions the absolute value of | x | which he got from the x^2 under the radical. if he were to use the absolute value for the x^2 why would he not use the absolute value of the lone | x | under the radical?(4 votes)
- Because the lone x was not a perfect square, he could not simplify the radical. Only if you are taking the principle root and you are SIMPLIFYING the radical can you put in the absolute value. If he would have put the absolute value sign for the x under the radical it would've become:

sqrt(|x|) = +/- (x^1/2)

Which still gives you the negative root which is extraneous for principle roots. Therefore putting the absolute value under the radical is never done when taking the principle root as it yields unwanted answers.(6 votes)

## Video transcript

What I want to do in
this video is resimplify this expression, 3 times the
principal root of 500 times x to the third, and take
into consideration some of the comments that
we got out on YouTube that actually give some
interesting perspective on how you could simplify this. So just as a quick review of
what we did in the last video, we said that this
is the same thing as 3 times the
principal root of 500. And I'm going to do it a
little bit different than I did in the last video, just
to make it interesting. This is 3 times
the principal root of 500 times the principal
root of x to the third. And 500-- we can
rewrite it, because 500 is not a perfect square. We can rewrite 500
as 100 times 5. Or even better, we could rewrite
that as 10 squared times 5. 10 squared is the
same thing as 100. So we can rewrite this first
part over here as 3 times the principal root
of 10 squared times 5 times the principal
root of x squared times x. That's the same thing
as x to the third. Now, the one thing I'm
going to do here-- actually, I won't talk about
it just yet, of how we're going to do it
differently than we did it in the last video. This radical right
here can be rewritten as-- so this is going to
be 3 times the square root, or the principal
root, I should say, of 10 squared times
the square root of 5. If we take the square root
of the product of two things, it's the same thing as taking
the square root of each of them and then taking the product. And so then this
over here is going to be times the square root
of, or the principal root of, x squared times the
principal root of x. And the principal root
of 10 squared is 10. And then what I said
in the last video is that the principal
root of x squared is going to be the absolute
value of x, just in case x itself is a negative number. And so then if you
simplify all of this, you get 3 times
10, which is 30-- and I'm just going to
switch the order here-- times the absolute value of x. And then you have
the square root of 5, or the principal root of 5,
times the principal root of x. And this is just going to be
equal to the principal root of 5x. Taking the square
root of something and multiplying that times the
square root of something else is the same thing as just
taking the square root of 5x. So all of this simplified
down to 30 times the absolute value of x times
the principal root of 5x. And this is what we
got in the last video. And the interesting
thing here is, if we assume we're only
dealing with real numbers, the domain of x right
over here, the x's that will make this
expression defined in the real numbers--
then x has to be greater than or equal to 0. So maybe I could
write it this way. The domain here is that
x is any real number greater than or equal to 0. And the reason
why I say that is, if you put a negative number
in here and you cube it, you're going to get
another negative number. And then at least
in the real numbers, you won't get an actual value. You'll get a square root
of a negative number here. So if you make this-- if
you assume this right here, we're dealing with
the real numbers. We're not dealing with
any complex numbers. When you open it up
to complex numbers, then you can expand the
domain more broadly. But if you're dealing
with real numbers, you can say that x is going to
be greater than or equal to 0. And then the absolute
value of x is just going to be x, because it's not
going to be a negative number. And so if we're assuming
that the domain of x is-- or if this expression is
going to be evaluatable, or it's going to have
a positive number, then this can be written as 30x
times the square root of 5x. If you had the situation
where we were dealing with complex
numbers-- and if you don't know what a complex number
is, or an imaginary number, don't worry too much about it. But if you were
dealing with those, then you would have to keep
the absolute value of x there. Because then this
would be defined for numbers that
are less than 0.