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### Course: 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

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# Worked example: rationalizing the denominator

Sal rationalizes the denominator of the expression (16+2x²)/(√8). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- In the last simplification, when everything is being simplified by two, why is the eight outside the square root (this eight->8 square root of 8) not simplified to a four?(22 votes)
- He didn't do that because the equation (8 * sqrt(2) + sqrt(2) * x^2) / 2 can be considered simplified. However, you could actually divide the 8 by 2 and get 4 * sqrt(2) + sqrt(2)/2 * x^2.(19 votes)

- why does he write √2x^2 and not x^2√2? (which to me looks like he wrote (√2)(x^2) instead of x^2√2)

im mildly confused why because i see x^2 as a whole number so is it just a matter of preference similar to writing (√2)(2) instead of 2√2 ?

i hope my question makes sense(15 votes)- Technically, you could write it the way Sal did, but it is much less confusing to write x^2sqrt2 so that is how it is usually written.(3 votes)

- I was asked this exercise during a challenge test: Rationalize the denominator of this expression:

(1+3√x)/(√(x-3)-1)

Is there anyone who kindly would explain how to rationalize this expression's denominator?

Thanks in advance(7 votes)- Sure. Let us look at the denominator only:

It has the pattern of (√a) − b

We multiply this by (√a) + b

[(√a) - b][ (√a) + b]

Using the FOIL method you probably learned toward the end of Algebra I:

(√a)(√a) + b√a - b√a - b(b)

= a - b²

Of course, to be able to do this, we have to multiply the numerator by (√a) + b, so that we don't change the value of the expression.

So, for your problem we would do the following

Denominator:

[ √(x−3) − 1 ][ √(x−3) + 1]

= (x−3) - 1

= x-4

But we also have to multiply the numerator by √(x-3) + 1]

(1+3√x)[√(x-3) + 1]

You can multiply that out if you want, but it can't really be simplified.

So the numerator is: (1+3√x)[√(x-3) + 1]

The denominator is: (x−4)(4 votes)

- What If there was a sum of two radicals? For example 1÷(√3 + √2)?(4 votes)
- Then to rationalize the denominator, you would multiply by the conjugate of the denominator over itself. The conjugate of a binomial has the same first term and the opposite second term. So you would multiply by (sqrt(3) - sqrt(2)) / (sqrt(3) - sqrt(2))(8 votes)

- why does he say principle square root of two instead of just the square root of two?(4 votes)
- The principle square root basically means the positive square root of two, as the square root could be positive or negative. Generally, when a problem asks you to find the square root of something or has some kind of radical, it means the principle square root. Note I said "generally".(7 votes)

- At about2:15Sal multiplies the fraction by the square root of 2 over the square root of 2. Isn't that the same as multiplying by one?(4 votes)
- Yes, it is exactly the same as multiplying by one. And since anything multiplied by one is unchanged, you can multiply by one any time you like. If you multiply by √2 / √2 then you don't change the value of the expression, but you introduce a factor of √2 on both the top and bottom of the fraction -- and this is helpful in removing a square-root from the denominator, which is the aim of this video.(4 votes)

- Do you use the same pattern to rationalize a numerator?(4 votes)
- You
*rarely*want to rationalise a numerator, but I've seen a couple of circumstances in which it is useful, and indeed these same techniques apply.(1 vote)

- Why can (1-(Sqr root)2)/-1 equal out to (sqr root) 2 -1?(4 votes)
- both of them are the same, just multipled by -1.

(1 - √2)/-1 = -1 + √2, is the same propety as 1/-1 = -1/1

What is the propety?

Every positive multipled by a negative number is a negative number

And every negative number multipled by a negative number is positive.

As we know dividing something is the same as multiplying by reciprocal of it, so the rule works for diving too.(2 votes)

- how do you do questions that has whole number times radical divided by whole number times radical. E.x 15(sqrt3) / 3(srqt8)(2 votes)
- Just rationalize the denominator, don't worry about the numerator. Thus:

15√3 ÷ (3√8)

First, simplify the radicals. √8 = (√4)(√2) = 2√2. Thus,

15√3 ÷ (3√8) = 15√3 ÷ (6√2)

Now simplify the integers (15/6) = (5/2). Thus,

= 5√3 ÷ (2√2)

Multiply numerator and denominator by √2

= 5(√3)(√2) ÷ (2√2)(√2)

= 5√6 ÷ (2*2)

= (5√6) / 4(3 votes)

- How do you subtract square roots in a denominator(2 votes)
- Kathie,

You can eliminate a square root from the denominator like this:

1 / √2

You multiply by √2 / √2

1 / √2 * √2 / √2Multiply the numerators together and denominators together.

(1*√2) / (√2*√2) Do the multiplication

√2 / 2

So if you had

3√2/4 - 1/√2

you multiply the second fraction by √2/√2 to get

(3√2)/4 - √2/2 Now find a common denominator

(3√2)/4 - (2√2)/4 Now you have a common denominator so you get

(3√2 - 2√2)/4 which is

1√2/4 or just

√2/4

I hope that is of help to you.(2 votes)

## Video transcript

We're asked to rationalize and simplify this expression right over here and like many problems there are multiple ways to do this. We could simplify a little bit then rationalize, then simplify a little bit more, or we could just rationalize and simplify. And just to make sure that you know what they are even talking about rationalize is just a fancy word, fancy way of saying we don't want to see any square roots of numbers in the denominator. Thats all it says. So try to get these things outside of the denominator. So the first thing we can do is just simplify then rationalize, then we can think about other ways to do it. So what I'd like to do first is say the principle square root of 8 that can be simplified a little bit because 8 is the same thing as the square root of 4 times 2 which is the same thing as the square root of 4 times the square root of 2. So we can rewrite this entire expression as, the numerator is still the same, 16 plus 2X squared. All of that over, we can rewrite this as the square root of 4 times the square root of 2. The principle square root of 4 we know is just 2. So the square root of 8 we can rewrite as 2 times the principle square root of two. And I've simplified a little bit, I've done no rationalizing just yet, and it looks like there is a little more simplification I can do first. Because everything in the numerator and everything in the denominator is divisible by 2. So lets divide the numerator by 2. So if you divide the numerator by 2, 16 divided by 2, or you could view it as multipying the numerator and denominator by one half. So 16 times one half is 8. 2X squared times one half is just X squared. And then 2 times the principle square root of 2 times one half is just the square root of 2. It is 1 square roots of 2. So this whole thing has simplified to 8 plus X squared, all of that over the square root of 2. And now lets rationalize this. The best way to get this radical out of the denominator is just multiply the numerator and the denominator by the principle square root of 2. So lets do that. So times the principle square root of 2 over the principle square root of 2. Now just to show that it works on the denominator what is the principle square root of 2 times the principle square root of 2? Well its going to be 2. And in our numerator, we are going to distribute this term onto both terms in this expression, so you have 8 times the principle square root of 2 plus the square root of 2 times X squared. And we could consider this done, we have simplified the expression, or if you want you could break it up. You could say this is the same thing as 8 square roots of 2 over 2, which is 4 square roots of 2, plus the square root of 2 times X squared over 2. So depending on your tastes, you might view this as more simple or this as more simple but both are equally valid. Now I said there were multiple ways to do this. We could have rationalized right from the get go. Let me start with our original problem. So our original problem was 16 + 2X squared, all of that over the principle square root of 8. We could have rationalized from the get go by multiplying the numerator and the denominator by the principle square root of 8. And so in our denominator we'll just get 8. And then in our numerator we would get 16 times the principle square root of 8, plus 2 times the principle square root of 8X squared. And now we could try and simplify this a little bit more. You can say, well, everything in the numerator and denominator is divisible by 2, so the 16 could become an 8 if you divide by 2. The 2 becomes a 1. And this 8 becomes a 4. And then you get 8 square roots of 8 plus the square root of 8X squared. All of this over 4. And then you say wait, this looks different from what we had over here. And the reason is we still haven't simplified this radical. We know that we can rewrite the principle square root of 8 as 2 square roots of 2. And then we can see again that everything in the numerator and the denominator is also divisible by 2, so lets do that again. So if you divide everything in the numerator by 2 you can get rid of this 2 and that 2. And everything in the denominator by 2, this will become a 2. So then you have 8 square roots of 2 plus the square root of 2X squared, all of that over 2. Which is exactly what we had gotten here.