Integrated Math 2–3 Honors — Radical Operations and Rationalizing Denominators
Corrected lesson transcript


## 0:00 — Radical Operations and Rationalizing Denominators


## 0:05 — Combining Like Radicals

[0:05] Neither of these two terms simplify.
[0:09] Are we able to combine these together?
[0:14] Yeah, are we able to add them?
[0:17] What do they add to?
[0:21] Six square root three.
[0:24] Why is that?
[0:24] They both have a factor of square root three.
[0:28] So we could factor that out.
[0:31] You get 2 plus 4 and that is 6 square root 3.
[0:38] The radicals, the way we add them is you add the coefficients and then multiply by what's
[0:46] You could factor out the x.
[0:47] Ok, these are called like radicals.
[0:52] Oops.
[1:00] Three square root two. Which other term is a like radical?
[1:05] Think of this as plus negative six square root two.
[1:13] Three square root two plus negative six square root two is negative three square root two.
[1:22] Four square root five plus seven square root five is eleven square root five.
[1:29] you're done the directions are either simplify or it might just say add okay

## 1:38 — Simplify Before You Combine

[1:38] These are not like radicals yet, but both radicals simplify.
[1:46] these simplify I think so so on the side we start simplifying eight is two four
[1:54] two two this is two square root two
[2:07] And this is 5 square root 2
[2:11] Now once we simplify and we see they happen to be like radicals so we can now combine them
[2:17] Two square root two plus five square root two is seven square root two.
[2:23] 2 plus 5 7 2
[2:35] So first we simplify all the radicals and then we combine all the like radicals.
[2:41] Makes sense? Okay. And subtracting, like I said, same thing as adding a negative so
[2:49] It's the same exact thing.

## 2:51 — Distribute Through Radical Expressions

[2:51] So now we're multiplying.
[2:57] If there's more than one term, terms or things separated by addition, what property do we
[3:02] have to use?
[3:06] What did you just say?
[3:08] Sorry.
[3:09] So here we're multiplying, but this has more than one term.
[3:15] Terms or things separated by addition.
[3:18] What property do we have to use? Distributive. What's the long name? Did we talk about that last week?
[3:30] Distributive property of multiplication over addition.
[3:34] So we distribute.
[3:50] And then that's two rad, square root three times square root three is square root nine.
[3:57] Square root three times square root six is square root eighteen.
[4:01] and then we just start simplifying
[4:07] 18 is
[4:17] Square root eighteen simplifies to three square root two.
[4:23] The result is six plus three square root two.
[4:27] We're at two plus six.

## 4:29 — A Multi-Step Radical Product

[4:37] You could first write two square root one hundred fifty, then simplify.
[4:43] at 150, but I noticed right, 5 was a factor, so I wrote it as just 5 times 5 times 6, and
[4:52] then that's a perfect square square root 25 is 5 just kind of saves you time a little bit
[5:02] okay questions on that question did I screw up sorry six times to square root 10 let's
[5:20] go 12 there we go thank you

## 5:23 — Multiplying Two Binomials

[5:23] Now we have two terms times two terms.
[5:30] What property are we going to have to use here?
[5:36] We are still using the distributive property of multiplication.
[5:42] But instead of distributing one term to each thing separated by the addition, we have to
[5:51] distribute the entire thing to each term.
[6:00] Distribute the entire first binomial to each term of the second binomial.
[6:19] So you distribute both terms, or the...
[6:24] And now once you distribute both terms each term here, okay now we have to use the distributive property again
[6:35] Right here teach turn
[6:43] So we got two times around five
[6:57] Two terms times two terms produce four terms before combining.
[7:03] and then we combine any like terms or like radicals.
[7:08] Is that wrong?
[7:24] And are any of these like radicals?
[7:27] No. So we're done.
[7:37] Where did this subtraction come from? That sign needs to be corrected.
[7:41] We have to distribute to both terms, keeping the negative with the four.
[7:51] So yes, this right here, we have to distribute to both of these.
[7:57] thank you for picking up on that that was bad I usually just leave the
[8:05] negative with the four and then just do plus then we definitely would not have
[8:08] Here we are multiplying.
[8:11] Again, we're going to distribute this whole thing to each term.
[8:34] And now we have to use the distribute property twice.
[9:00] Oh
[9:30] And this guy simplifies right.
[9:49] And none of these are like radicals, so we're all done.
[9:57] Again this is all addition, so the order of any of those terms does not matter.
[10:01] Oh, same thing.

## 10:03 — Conjugates and Difference of Squares

[10:03] I'll just review this first one.
[11:09] The middle terms add to zero, leaving two minus sixteen, or negative fourteen.
[11:11] zero zero get negative 14
[11:42] Let's try another one.
[11:56] Try this one on your own.
[13:55] The middle terms add to zero, leaving nine minus twenty-eight, or negative nineteen.
[13:57] I get 9, 7, almost 28, middle 19.
[14:29] Both conjugate products simplify to integers.
[14:37] Why is that?
[14:43] It's like a difference of squares.
[14:48] Yeah, we're going to talk about that in a minute.
[14:50] The only difference between these two sets of two terms is one has a minus one has a
[14:58] plus.
[15:02] Same thing with this one.
[15:05] Rad 2 plus 4, square root 2 minus 4.
[15:10] Whenever you have that they are called conjugates.
[15:14] When you multiply two conjugates, you always end up with the middle two terms end up adding
[15:23] to be zero.
[15:25] Okay, and we're going to need that later on.
[15:28] uh you know define a conjugate for you later

## 15:35 — Rationalizing a One-Term Denominator

[15:35] That covered adding, subtracting, and multiplying. Now we are dividing radicals.
[15:42] How are we going to divide?
[16:09] Here's what we're going to do.
[16:10] This process is called rationalizing the denominator.
[16:26] What's the definition of a rational number?
[16:29] A rational number can be written as a ratio of two integers.
[16:41] a perfect square is an irrational number.
[16:45] It cannot be written as a fraction.
[16:47] So what we're going to do is we're going to rationalize the denominator.
[16:52] to turn the denominator into a rational number.
[16:57] And here's how we do it.
[17:03] We want to get rid of this square root.
[17:06] Multiply by square root two over square root two, which equals one.
[17:12] We can't just multiply by square root 2.
[17:15] But you can multiply by square root 2 over square root 2.
[17:18] Why can we do that?
[17:18] It's just number one.
[17:23] Okay.
[17:24] The denominator becomes four and the numerator becomes square root twelve.
[17:31] over two times two.
[17:38] So it's square root 12 over four.
[17:44] And then we simplify any radicals.
[17:53] Square root twelve is two square root three, so the fraction simplifies to square root three over two.
[18:01] And then the last thing we do is simplify.
[18:07] 2 goes into the 1s.
[18:10] Rad 3 over 2.
[18:15] So we now rationalize the denominator.
[18:17] The denominator is a rational number.
[18:22] Okay, and this is basically how we're going to divide.
[18:26] All we do is rationalize the denominator
[18:30] And again, the direction might say divide or might just say simplified.
[18:39] And so what am I going to multiply up here?
[18:41] Rad 3 over square root 3.
[18:48] Multiply by square root three over square root three.
[18:55] So square root 45.
[19:01] And what is square root 45?
[19:13] Square root forty-five is three square root five, so the result is square root five.
[19:16] And this is where I left.
[19:17] Multiply by square root five over square root five.
[19:19] I'm going to do it.
[19:22] Red five by the square root five.
[20:02] The expression simplifies to two square root two over three.
[20:19] We've got questions on that.
[20:22] Again, we rationalized the denominator.
[20:23] This is not a rational number.
[20:25] So no, we don't want any square roots in the denominator.
[20:30] Let's try another one.

## 20:31 — Rationalizing with a Conjugate

[20:31] Do not multiply this two-term denominator by square root five alone.
[20:34] We're not going to multiply by square root 5.
[20:37] Because if you do, we get 3 square root 5 plus 5.
[20:45] We did get rid of this square root.
[20:49] But now we got another one stuck here.
[20:54] Multiply by square root 5 again.
[20:56] Yeah, you should try that.
[20:58] Try that.
[21:02] Now we're at 2 times 5 is 10 over 3 times 5 plus 5 right 5.
[21:10] Wait, one more time.
[21:11] One more time?
[21:12] Yeah.
[21:15] Okay, so that doesn't work.
[21:24] What did somebody say over there?
[21:28] Multiply by the conjugate of the denominator.
[21:36] The conjugate of three plus square root five is three minus square root five.
[21:43] So we're going to multiply by 3 minus square root 5 or 3 minus square root 5.
[21:58] Now we've got a lot of distributing.
[22:06] 6 minus 2 square root 5.
[23:00] And once you multiply by the conjugate, the two middle terms always has to be zero.
[23:11] After multiplying by the conjugate, the two middle denominator terms cancel.
[23:22] And we're almost done.
[23:25] Why are we not quite done?
[23:27] We definitely rationalized the denominator.
[23:31] But they all have a factor of 2 in it.
[23:38] Okay, you could do this.
[23:39] 6 over 4 minus 2 square root 5 over 4.
[23:44] Which is 3 over 2 minus
[23:48] square root 5 over 2.
[23:52] Factor two from the numerator and simplify to three minus square root five over two.
[24:00] The other option is you just factor out a 2.
[24:07] We get 2 times 3 minus square root phi over 4.
[24:19] And that simplifies the one.

## 24:20 — A Larger Conjugate Example

[24:20] okay questions on that so if we're dividing by something with two terms as
[24:31] As a radical we have to multiply by the conjugate over the conjugate of the denominator
[24:38] Let's try this one
[25:12] Are we going to multiply by the conjugate of the numerator or denominator?
[25:17] Denominator.
[25:18] Our whole goal is to get rid of square roots in the denominator.
[25:21] So we're going to multiply by 3.
[25:24] You just switch the sign.
[25:25] plus square root 3 plus square root 7. And now we have lots of distributing to do.
[26:48] Distribute carefully through the numerator and denominator.
[26:50] Don't mess up.
[26:55] plus four and six, plus seven, plus 14.
[27:10] It becomes three.
[27:48] Are any of the ones in the numerator like radicals?
[27:55] no that's gonna be it right there it's the best we could do okay what

## 28:09 — Lesson Summary

[28:09] Adding and subtracting use like radicals; multiplying uses distribution; dividing requires rationalizing the denominator.
[28:15] dividing divided is the only weird one we don't really divide we multiply by
[28:19] something to rationalize the denominator when there's two terms you multiply by
[28:27] the conjugate so a plus B the conjugate
[28:40] is A minus B and vice versa. These are conjugates. All we do is switch the term.