Integrated Math 2–3 Honors — Laws of Exponents and Simplifying Expressions Corrected lesson transcript ## 0:00 — Laws of Exponents ## 0:05 — Repeated Addition and Multiplication [0:05] Okay, going back to third grade. [0:13] Maybe second grade. [0:19] What does 3 times 4 mean? [0:22] Not what it is. [0:24] Three plus three plus three plus three. [0:27] Three plus three plus three. [0:30] Four plus three. [0:31] Four plus three. [0:32] Four plus three. [0:32] Four plus three. [0:33] What does that mean in general? [0:34] Four plus four plus four. [0:36] 4 plus 4. [0:40] Multiplication is just repeated. [0:43] It is. [0:44] Okay. [0:46] Just for fun, what does x times y mean? [0:51] x plus x plus x, repeated y times. [1:03] How many times? [1:04] Y times. [1:07] Hence the word times, right? [1:14] Or it could mean, [1:17] Y plus Y plus Y. [1:23] Next time. [1:26] Now we're going to, I don't know, fifth grade. [1:31] Fifth grade. [1:32] Maybe like 8th grade? [1:34] 8th grade? [1:35] Maybe like 10th grade. [1:37] Are you sure? [1:38] What does 3 to the power of 4 mean? [1:41] 3 times 3 times 3 times 3. [1:48] 4? [1:50] Uh... [1:51] That's it. There is no more. [1:53] Okay. [1:56] Exponents is just repeated? [1:59] Multiplication. [2:00] Multiplication. [2:01] Okay. Because of that, we got lots of rules. Or laws, whatever you want to call them. ## 2:13 — Product Rule for Exponents [2:13] Did you guys learn exponent laws last year? [2:21] Sorta. [2:23] Sorta. [2:26] How could I rewrite this? Three to the fifth times three to the fourth is three to the ninth. [2:37] Why is that? Because five plus four is nine. [2:41] Right, so 3 to the 5th is 3 times 3 times 3 times 3 times 3. [2:49] This is the same multiplication times 3 times 3 times 3 times 3 [2:59] Okay, which is 3 to the 9th [3:06] It's 3 to the 5th times 3 to the 4th is 3 to the 5 plus 4 [3:16] Which is 3 to the 9th, okay [3:19] Our first law is x to the a times x to the b equals x to the a plus b. [3:40] Very important. [3:43] They must have the same base, right? [3:50] If it was X to the A times Y to the B [3:54] We could do nothing with it so they got to have the same base [4:01] I [4:02] Don't know if I mentioned this but [4:07] This is called the base [4:13] This value is called the exponent, or the power. [4:19] Or we call it the power. [4:27] The way we say it would be x to the 8th power, [4:37] or just x to the 8th, x to the 8th, [4:43] When the exponent is 2, we also say x squared. Do not say x 2. [5:03] say this x to the power of 2 when there happens to be a 2 we also call it x [5:11] squared squared don't ever say x 2 ok it's x to the second power x to the [5:46] The directions are to simplify, not solve, because this is an expression rather than an equation. [5:53] Solve? [5:54] Simply multiply. [5:55] Definitely not be solved. [5:56] Why would it not be solved? [5:57] It's not an equation. [5:59] It's not a equation. [6:00] We don't solve expressions. [6:02] it could be simplified typically. Okay and what are we gonna do first? Okay this is [6:17] important. Terms in algebra are things separated by addition. There's only one [6:25] term here that whole thing is a term okay so we're not combining like terms [6:33] we're just simplifying it okay before we use this law what do we need to do first [6:45] Put the factors with the same bases next to each other. [6:51] Rewrite the product as x to the fifth times x cubed times y to the fourth times y to the tenth. [7:00] And what law or what property allows us to just switch the order of this? [7:08] Commutative. [7:09] Commutative property of? [7:13] Multiplication. [7:13] Multiplication. [7:15] OK. [7:19] We could switch the order of multiplication by the commutative property. [7:24] The result is x to the eighth times y to the fourteenth. [7:42] Okay, when you're doing these, do we need to show this step? [7:48] Not really. [7:49] No. [7:50] We can do that in our head. [7:52] Do we need to show this step? [7:53] No. [7:54] Not really. [7:55] We will need to show this step in cases like x to the a times x to the fifth. [8:07] Rewrite this as x to the a plus 5. [8:09] 8 plus 5. [8:10] 8 plus 5. [8:11] There's no other way to write that. [8:14] OK, so definitely in those cases, [8:17] we have a variable there instead of just a number. [8:20] But otherwise, you can skip this step, skip this step. [8:24] And a lot of times, just do it right in your head, [8:26] just like that. ## 8:40 — Distribute, Then Combine Bases [8:40] What do we do first here? [8:44] Okay, whenever we have addition, okay, to use the laws we just learned, it has to be [8:53] all multiplication. [8:54] So we do need to use the distributive property. [8:57] What's the real name of the distributive property? [8:59] Or the full name I should say. [9:05] The distributive property of multiplication, this is the property of this multiplication [9:11] right here, over addition. [9:14] So it's the distributive property of multiplication over addition. [9:18] So we distribute over that addition. [9:35] And [9:40] And now we use the law. [9:43] You can rearrange everything. [9:45] You don't have to. [9:48] Multiply the coefficients and add exponents on like bases: 6x to the eleventh y squared plus 8x to the fifth y cubed. ## 9:58 — Quotient Rule for Exponents [9:58] And can we add in? [10:03] No. [10:04] No, these are not like terms. [10:07] Like terms must have the same variables raised to the same powers. [10:11] variables to the same exact powers. [10:16] Otherwise you can't combine them by adding them. [10:20] They're not like terms. [10:36] Now we are dividing two powers with the same base. [10:45] What is this going to be? [10:49] Three to the sixth divided by three to the fourth is three to the six minus four, or three squared. [10:51] 3 to the power of 6 minus 4. [10:56] Which is 3 to the power of 2. [11:01] Why is that? Why are we going to subtract? [11:06] If the multiplication has a win dividing the opposite to use in fact. [11:12] Oh yeah, that would logically make sense. But why is that? [11:21] Four of the factors cancel, leaving two factors of 3 in the numerator. [11:40] I like that you don't really need to write this step [11:50] The quotient rule is x to the a divided by x to the b equals x to the a minus b. [12:02] one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, [12:02] one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, [12:06] divided by x to the b. When you solve problems like this you just leave them in their exponent [12:14] form right you don't like simplify further. You can't you mean multiply it out yeah because it's [12:19] just a number all right there's no like hard and fast rule you could multiply it out you could just [12:25] leave it okay um I mean you definitely could write this as nine most of our expressions are contained [12:34] variables so either way we're practicing the rules there's no right this is the [12:42] same thing as this just written in two different ways okay so the lot X to the [12:49] A divided by X to the B as long as they have the same base now we no longer have [12:55] a denominator. ## 13:02 — Negative Exponents [13:02] By the quotient rule, x to the fourth divided by x to the sixth becomes x to the negative second power. [13:14] what in the world a negative we can and what does this even mean like that why [13:33] is that that's true but why is that it's like you have to move it down to make it [13:43] positive. But I'm there now. [13:46] No, because like, there's like, like, I know, like in practice, [13:49] I can like move it down. [13:51] Just move stuff down. [13:54] Right? Isn't that how it works? [13:58] Sort of. Yes. [14:02] I like that. So [14:17] After common factors cancel, we have 1 divided by x times x, or 1 over x squared. [14:29] The negative-exponent rule is x to the negative a equals 1 divided by x to the positive a. [14:38] x to the negative a is 1 divided by x to the positive a. [14:51] Very important. ## 14:58 — Simplifying Rational Monomials [14:58] Assume none of the variables are zero. [15:05] That's important because we can't necessarily simplify but we're going to assume they're [15:11] never zero. [15:14] So what now? [15:22] Well we could break it up like this. [15:26] We're not going to. [15:28] We're going to do this part in our head. [15:38] And then we could use that law for each one individually. [15:48] Using the quotient rule gives one fourth times x to the three minus one times y to the five minus eight. [16:07] Okay, and we're typically not going to leave anything with negative exponents. [16:13] So we don't want this as our final answer. [16:15] We need to get rid of the negative exponent. [16:17] This becomes one fourth times x squared times 1 over y cubed. [16:32] The final simplified form is x squared over 4y cubed. [16:39] get that that would be our final simplified form okay just kind of a [16:57] shortcut where we don't have to go through all these steps will be as a [17:05] know shortcuts. Just like we do like this, we rarely write this as 2 over 2 times 4, [17:22] 2 over 2 times 1 over 4. 2 over 2 is 1. [17:33] What do we normally do? [17:36] Reduce 2 over 8 to one fourth, then cancel common variable factors. [17:39] And we get a 2. [17:41] And we have 4. [17:43] And you just write 1 fourth. [17:45] So we're going to do the same thing with variables as well. [17:47] So first we can do that. [17:48] And this only works if everything is multiplication. There's addition. Don't ever do this [17:54] Okay, so we just go one [17:57] fourth [18:00] Cancel one x factor to leave x squared, and cancel five y factors to leave y cubed in the denominator. [18:06] Five of these simplify with five of these and we get y cubed [18:13] 1 times x squared times 1 is x squared over 4 times 1 times y cubed 4y cubed. [18:23] When numerator and denominator are products, simplify variable factors just as you simplify fractions. [18:30] simplify just like we simplify fractions. [18:35] Okay, you can do either way. [18:39] But it's usually fastest if we do it this way. [18:46] Okay. ## 18:47 — Powers of Products and Quotients [18:47] Now we have a product raised to a power. [18:59] If I wrote it out, it would be what? [19:01] 3x times 3x times 3x times 3x. [19:06] Agree? And then if you want, you could rearrange it. [19:14] And then this is 3 to the fourth x to the fourth. [19:24] The power-of-a-product rule is open parenthesis xy close parenthesis to the a equals x to the a times y to the a. [19:36] become? [19:36] X to the A, don't say XA. Y to the A, or Y to the A power. [19:51] This is when we have a product inside an exponent. [19:57] You raise every one of the things inside to the power of whatever that exponent is. [20:08] And we might as well do one more before we do some more examples. [20:26] A power raised to a power multiplies the exponents. [20:29] The entire thing is raised to an exponent. [20:34] We got, if I wrote it out, it's 3 squared times 3 squared times 3 squared times 3 squared. [20:45] Now they all have the same base. [20:48] So it's 3 to the 2 times 2 times 2. [20:51] Plus 2. [20:53] Plus 2. [20:54] Plus 2. [20:54] plus 2, which is 3 to the 2 times 4 repeated addition. [21:06] Which we end up just multiplying these exponents. [21:09] The power-to-a-power rule is open parenthesis x to the a close parenthesis to the b equals x to the ab. [21:13] So our next law is x to the a to the power of b is x to the a times b. [21:46] And actually I meant one more. [21:52] The power-of-a-quotient rule is open parenthesis x over y close parenthesis to the a equals x to the a over y to the a. [22:03] So if you have a quotient inside an exponent, the numerator gets the power and the denominator ## 22:09 — A Multi-Step Power Example [22:09] Okay, let's do some examples. [22:25] Yes. [23:04] Apply the outer third power to the numerator and denominator. [23:08] What rule are we going to use first? [23:16] There's no addition, we're definitely not distributing anything. [23:21] So, overall we have a quotient, okay, inside the exponent. [23:29] This becomes open parenthesis 3x squared y cubed close parenthesis cubed over open parenthesis 2z to the fourth close parenthesis cubed. [23:36] So it becomes 3x squared y cubed to the third power, 2z to the fourth to the third power. [23:50] And now within the numerator and within the denominator, [23:53] we have a product to a power, a product to a power. [24:03] Apply the third power to every factor in the numerator and denominator. [24:07] X squared to the power of three, [24:11] Y cubed to the power of three, [24:15] two to the power of three, [24:17] Z to the fourth power of three. [24:32] Multiply the exponents in each power-to-a-power expression. [24:38] a power. [24:39] Base exponent raised to a power. [24:43] And we multiply those. [24:46] So it's 3 cubed x squared to the 2 times 3, y to the 3 times 3, 4 times 3. [25:01] You can't just do that multiplication in your head. [25:03] You don't have to write that part out. [25:20] You could multiply out the 3's. [25:24] The simplified result is 27x to the sixth y to the ninth over 8z to the twelfth. [25:36] And then do we need to simplify anything? [25:40] No, these are, the coefficients are not going to reduce and they don't have any of the same [25:48] variables so there's no more simplification. [25:50] This would be it. [26:08] questions on that? [26:10] are we done? [26:14] not yet, not yet, not yet. ## 26:20 — Negative-Exponent Practice [26:20] Let's do a couple more. [26:22] Simplify each base using the quotient rule. [26:38] For x, subtracting a negative exponent gives 4 minus negative 5, or 9. [26:44] is x to the 9. The other way we could do this is to use the [26:52] Equivalently, x to the negative fifth in the denominator becomes x to the fifth in the numerator. [27:03] multiplied by the reciprocal of the denominator and x to the fourth times x [27:12] fifth is x to the ninth. Or we could just do this three times x to the four minus negative [27:26] The expression becomes 3x to the ninth times y to the negative first. [27:45] With positive exponents, the final answer is 3x to the ninth over y. ## 27:53 — The Zero-Exponent Rule [27:53] Oh wait, I forgot one major thing. [28:04] What does a zero exponent mean? [28:08] One, why is that? [28:12] I was told. [28:13] You were told? [28:14] Yeah. [28:15] It's correct, but what would be a reason why this would be up? [28:19] When you go up, my exclamation multiply, so if you go down, you divide. [28:24] So much of our 1 is 5, and then you divide by 5 and you get 1. [28:29] OK, I like that. Let's go with 125. [28:37] 25, 5, or this is what, 5 cubed, [28:41] 5 squared, 5 to the first, [28:45] 5 to the 0. You divide, [28:49] you divide, you divide, this becomes 1. Divide by 5. [28:52] OK, anything to the power of zero is one, except zero to the zero is undefined. [29:12] But besides that, anything to the power of zero is one. [29:17] It's very important. [29:22] And if you ever end up with like x to the fourth divided by x to the fourth, obviously [29:29] this simplifies to just one, right? [29:32] But if you use our rules, we have four minus four, which is x to the zero, and the zero [29:39] is one.