Integrated Math 2–3 Honors — Rational Exponents and Simplifying Radicals
Corrected lesson transcript


## 0:00 — Rational Exponents and Radicals


## 0:05 — Fractional Powers Are Roots

[0:05] The other option is let's kind of check this out.
[0:07] 49 is the same thing as seven squared, right?
[0:12] Okay, and then using our laws of exponents, a base to a power raised to a power, we multiply
[0:20] them, and it's 7.
[0:25] Let's try another one for fun.
[0:33] I could write 25 as what? 5 squared, which is 5 to the 2 times one half, which is 5.
[0:47] Okay, which is the same thing as the square root of 25.
[0:55] Uh, let's try a different one.
[1:10] What number squared is 30?
[1:14] It's not a perfect square, but there is a number, when you square it is 30 and it is.
[1:23] It's the square root of 30. That's the definition of a number that when you square it equals
[1:30] that number. The definition is the square root.
[1:36] And that is 2 times 1 half, which is the square root of 30 to the power of 1, which is the
[1:43] square root of 30.
[1:44] Okay, so x to the 1 half power is exactly the same thing as the square root of x.

## 1:51 — Cube Roots and Higher Roots

[1:51] That's the one-half power.
[1:52] What about a one-third power?
[1:54] Why is that?
[2:01] What number cubed is 8?
[2:05] Two.
[2:08] Oops.
[2:11] Which is two to the three times one third.
[2:14] Which is two to the first, which is two.
[2:18] Okay.
[2:18] which is the cube root of 8. The number that when you cube it is 8 is the cube root of 8, which is 2.
[2:33] What about a number that doesn't... let's do this... 10 to the one third.
[2:40] now. The number that when you cube it is 10 is by definition the cube root of 10.
[2:53] And this becomes the cube root of 10. So X to the 1 third is exactly the cube
[3:03] root of x. Is that going to work with every single denominator? One fourth? One fifth?
[3:14] One sixth? Yes. Okay, like,
[3:30] 81 to the one-fourth.
[3:39] What to the power of four is 81?
[3:43] Three.
[3:49] which is the same thing as the fourth root of 81.
[3:57] X to the power of one over any number b is the bth root of that number.

## 4:05 — Evaluating Rational Exponents

[4:05] Uh oh, now it's not just one over a number in the next one.
[4:12] I have 64 to the two-thirds power.
[4:22] Is this possible yeah, how are we gonna do this or deal with this?
[4:36] Okay, so one of our laws
[4:40] 2 thirds is the same thing as 1 third times 2.
[4:46] Well, that's not a law; that's just multiplication.
[4:50] One of our laws is x to the a to the power of b is equal to x to the a times b.
[5:01] So here I got this part right here.
[5:05] So I'm going to write this as 64 to the one third to the power of 2.
[5:13] We know the one third power is of 64 is 4.
[5:23] And we have 4 squared is 16.
[5:31] And we could have done it this way as well.
[5:38] 64 to the 2 times 1 third, which is 64 squared to the 1 third.
[5:51] What is 64 squared?
[5:53] Anybody have a calculator?
[5:57] It said 4,096.
[6:02] Oh yeah.
[6:03] Somebody do it on the calculator.
[6:07] I think it's 4,096.
[6:13] I got it.
[6:14] 4,096.
[6:16] And then what is the cube root of 4,096?
[6:23] Yeah, it better be exactly 16 where math is dead if that's not the case
[6:33] Okay notice it's a lot easier to first do the root
[6:39] Because it makes a small number and then we do the power a
[6:47] Lot easier than making the number a lot bigger and then who knows what the cube root of that number is
[6:53] OK, that gets difficult.
[6:56] So in general, x to the a over b can be written two different ways.
[7:06] It could be the b-th root of x, raised to the power of a.
[7:17] Or the b-th root of x to the power of a.

## 7:20 — A Fifth-Root Example

[7:20] Let's try this one.
[7:30] So that's going to be the what root?
[7:38] Fifth root.
[7:39] Whatever the denominator is, is the root.
[7:44] To the power of three.
[7:48] What is the fifth root of 32?
[7:52] Two.
[7:53] Two cubed is 8. The other way to do it is to take our base, which is
[8:08] 32 and just start factoring it.
[8:16] 2 to the 4, 2 to the 2.
[8:21] So that's 2 to the 5th.
[8:27] Just
[8:27] keep it as exponents, don't mess with anything. A base with an exponent raised to an exponent
[8:34] that's 2 to the 5 times 3 fifths and what is 5 times 3 fifths 2 cubed which is 8 so
[8:47] you leave it all as exponents once you factor the whatever the base is and then
[8:54] to just use the rules of exponents to multiply them.

## 8:57 — Principal Roots and Radical Vocabulary

[8:57] The square root of 49 is?
[8:58] Seven.
[8:59] Seven.
[9:01] Is it also negative 7?
[9:03] OK, this is very important.
[9:05] Let's talk about a few things.
[9:10] First up, is there a, what's the root here?
[9:14] 2.
[9:15] Or the, OK, that's called the index, or the indices index.
[9:22] Tell you the root.
[9:23] Square root obviously has an invisible 2 there.
[9:27] This is called the radical symbol.
[9:34] Radical sign. Okay.
[9:36] And the thing inside it.
[9:38] It's called the radicand.
[9:45] Okay. Whenever you see just the radical symbol,
[9:53] that is only the positive square root.
[9:55] This is only 7. 49 does have two square roots, positive and negative 7.
[10:06] But if you see the radical sign, this is only the positive.
[10:10] What is this equal to?
[10:13] And sometimes we just say rad 24. This really means the square root
[10:34] of 24. We often shorten it and say rad. Is 24 a perfect square? No. Between what
[10:45] two integers is this oh definitely between four and five it's gonna be four
[10:52] point something

## 10:59 — Simplifying Square Roots

[10:59] The only way to write exactly the square root of 24
[11:03] is to just leave it like this.
[11:06] However, we can simplify this.
[11:11] And here's what we do.
[11:14] What perfect square goes into 24?
[11:20] 2 is not a perfect square.
[11:22] 4 is a perfect square. So this factors to 4 times 6. We agree? Okay. And then one of
[11:37] our very important rules, the square root of a times b is equal to the square root of
[11:45] a times square root of b. Why is that true? Or is that true?
[12:03] Let's look at this. Square root of a times b is the same thing as a times b to what power?
[12:13] One half. And when we have a product raised to an exponent, what can I do with this?
[12:23] Raise each factor, a and b, to the one-half power: a to the one half times b to the one half.
[12:30] Which is one of the laws of exponents we talked about Tuesday, which is the square root of a times the square root of b.
[12:38] OK, so that's why this guy is true.
[12:44] So we're going to use that.
[12:46] The square root of 4 times 6 is the square root of 4 times the square root of 6.
[12:52] And the square root of 4 is 2.
[12:57] OK, so even though this cannot be written as a whole number or a decimal or anything,
[13:04] We still simplify it to this
[13:11] Okay

## 13:12 — Prime Factorization and Factor Pairs

[13:12] So 50 is not a perfect square, but can we simplify it?
[13:16] What perfect square goes into 50?
[13:26] 25 times 2, which I could write as rad 25 times rad 2, which is 5 rad 2.
[13:41] Five times square root of 2.
[13:47] There's a slightly easier way than trying to figure out what perfect square goes into
[13:51] it.
[13:53] You just take this radicand and you just start factoring it to its prime factorization.
[14:03] Anytime you have a pair here, it's going to be a perfect square.
[14:12] So I could write this as...
[14:19] 5 squared times 2, which we usually don't write this.
[14:30] The square root of 5 squared is 5.
[14:35] So typically what I do is I factor it.
[14:41] I circle the factor pairs.
[14:45] And I know it's going to be 5, rad, whatever is left over.
[14:50] 2.
[14:51] And you're done.
[14:51] it's not obvious what the perfect square goes into this
[14:54] or it might be to you, but just start factoring it. 72 is what?
[14:58] 2 times 36. 2 times
[15:05] Two times
[15:07] 9. 3 times 3
[15:13] Every pair is a perfect square.
[15:19] So it's going to be 2 times 3 times whatever is left over.
[15:26] Rad 2.
[15:28] So it's 6 rad 2.
[15:29] Try this one on your own.
[15:33] So, it's going to be 2 times 2 times rad 2 times 3, 4 rad 6.
[15:52] If you want to rewrite some people like this, 2 times 2 times 2 times 2 times 2 times 3,
[16:02] Every pair is a perfect square. This becomes 2 times 2 times the square root of 2 times 3.
[16:11] Or you can just do that part in your head.

## 16:22 — Cube Roots: Groups of Three

[16:22] Is 40 a perfect cube?
[16:45] Instead of thinking what perfect squares go into this sorry perfect cubes just start factoring
[16:52] Two.
[16:55] Two, two.
[16:57] 10 to
[17:00] 5
[17:02] Now are we looking for pairs?
[17:06] Yeah, we need three of a kind because when we have three they multiply to be a perfect
[17:14] cube.
[17:18] So now this is going to be two times the cube root of five.
[17:26] You would not say two rad five because rad typically just means square root.
[17:33] 2 times the cube root of 5.

## 17:37 — Fourth Roots: Groups of Four

[17:37] Just start factoring: 2 times 80; 2 times 40;
[17:52] 2 times 20; 2 times 10; 2 times 5.
[17:57] Now how many do we need?
[17:59] 4 We need 4 of a kind
[18:04] Which is 16 So it's going to be 2 times the 4th root of
[18:11] 2 times 5.

## 18:12 — Variables Under a Radical

[18:12] We're going to assume all the variables are positive.
[18:21] Now we got numbers and variables.
[18:25] Uh oh, what are we going to do there?
[18:31] Let's deal with them separately.
[18:33] So on the side, let's factor the 20.
[18:42] We've got a pair there.
[18:46] And if you factor x cubed, it's x times x times x.
[18:55] We've got a pair here.
[18:57] That's x squared.
[18:59] So, on the outside it's going to be 2 times x, and then whatever is left over, which is
[19:08] a 5 and an x.
[19:12] So it's 2x times rad 5x.
[19:14] Let's try another one.
[19:23] So how many x's on the outside?
[19:35] So it's going to be 3x to the power of 3 times the square root of 2x.

## 19:49 — Cube Roots with Variables

[19:49] Now we're going to simplify not the square roots, but cube roots.
[19:56] So, how many are we looking for at a time?
[20:03] Three of a kind.
[20:08] So we definitely have a 3 on the outside. How many x's are we looking for?
[20:16] 3. So we got an x. And then we're left over with a 3 and an x squared.
[20:25] That would be it.
[20:32] Is this making sense?
[20:35] Yes.
[20:36] Okay.

## 20:36 — Large Exponents and Remainders

[20:36] okay we're gonna write out 43 X's can circle every three of them no we need a
[20:43] So yeah, we're going to have exactly 14 x cubes
[20:48] And a remainder of one more
[20:52] So it's going to be X to the 14 times the cube root of X
[21:00] Another option is to do this
[21:04] What was the X to the 43
[21:08] Let's just change this to X to the 43 to the 1 3rd
[21:16] Which is X to the 43 times 1 3rd
[21:20] Which mixed numbers are terrible, but
[21:26] 3 goes in the 43 14 times
[21:31] 14 and 1 third
[21:38] But what does fourteen and one third mean?
[21:44] That means 14 plus one third.
[21:49] Mixed numbers are horrible.
[21:53] In algebra, when there is no symbol between factors, it means multiplication.
[22:01] Mixed numbers really mean addition.
[22:02] Okay, which is x to the 14th times x to the one-third.
[22:07] Which is x to the 14th times the cube root of x.
[22:15] Same idea of just dividing the exponent by the index and whatever the remainder is, that's
[22:21] how many are up here.

## 22:23 — A Multi-Variable Fourth Root

[22:23] Okay, don't start factoring 7,000.
[22:30] Is there a perfect power of 4 that goes in 7,000?
[22:34] Sorry, 70,000?
[22:43] Four zeros is ten to the power of four.
[22:52] So let's rewrite this as seven times ten to the power of four.
[23:05] And sometimes we could do well
[23:12] Four goes into four how many times
[23:16] Once okay, so this is gonna go on the outside well
[23:21] Let me do this
[23:23] We got seven out here right no we got a ten out here
[23:28] We got seven in here
[23:32] This is a perfect power of 4.
[23:35] So the 4th root of x to the 4th is an x.
[23:38] 4 goes into 7 how many times?
[23:42] Once.
[23:44] And how many is left on the inside?
[23:46] 3.
[23:50] 4 goes into 10 twice.
[23:54] How many is left over?
[23:57] 2.
[23:57] This will be z squared.