01 - Simplify Rational Exponents (Fractional Exponents, Powers & Radicals) - Part 1 - Free Educational videos for Students in K-12

01 - Simplify Rational Exponents (Fractional Exponents, Powers & Radicals) - Part 1 - By Math and Science

Transcript


00:00	Hello , welcome back to algebra . This is actually
00:02	a new unit of algebra where the end game is
00:05	going to be for us to learn about the very
00:07	important concept called the exponential function and also a related
00:11	function called al algorithm . Probably most people have heard
00:14	of the term exponential and log algorithm . We're gonna
00:16	be culminating this unit of lessons in studying those extremely
00:21	important functions . But here in this lesson we're going
00:23	to start off the discussion by talking about something called
00:26	rational exponents . Of the title of this lesson is
00:29	called rational exponents . Another way to say it is
00:32	exponents that contain fractions fractional exponents . Now I want
00:35	to stay up in the up in the beginning that
00:37	we have studied this in in some degree in the
00:40	past when we talked about radicals , cubed roots ,
00:43	fourth roots and things like that . We talked about
00:45	fractional exponents , rational exponents here we're going into much
00:49	more depth with much more kind of complex problems .
00:53	To prepare us to study the concepts of the exponential
00:55	function and logarithms , which are coming up very very
00:58	soon . So we have to review a couple of
01:00	things to make sure everybody is on the same page
01:02	before we get going . So we uh we can
01:05	recall the following things . These are things uh that
01:08	you all should know from previous lessons and we're gonna
01:12	start very basic , we know that five square ,
01:14	that's an exponent , right ? And we know that
01:16	that's five times five , we know that's equal to
01:18	25 so far so good . Not too hard .
01:22	We've also learned about negative exponents . So here's a
01:25	positive exponent . A negative exponent might be five to
01:27	the negative two . And we learned that when we
01:29	have a negative exponents , all you do is you
01:31	drop that guy below a fraction and make the exponent
01:34	positive . We talked about all the reasons why this
01:37	is the case in the past . So if something
01:39	like this looks foreign to you , you need to
01:41	go back to the more basic lessons on negative exponents
01:45	. When you have a negative exponent you drop it
01:46	down , make the exponent positive , which means it
01:48	becomes 1/5 times 5 , 1/25 . Okay . And
01:53	then we also learned the very important exponents . When
01:56	we discussed radicals and exponents a long time ago ,
01:58	if we have an exponent , any number raised to
02:01	the zero as an exponent it is by definition equal
02:04	to the number one . Now again , if this
02:06	looks weird to you , if you've never seen it
02:08	before , go back and look at the more basic
02:10	lessons and experience . We've talked extensively why raising something
02:13	to the zero power actually is defined to be one
02:16	in math . Now what we're doing in this lesson
02:19	is we're going a little bit beyond these basic ideas
02:22	and we're talking about rational exponents , which means fractional
02:25	exponents exponents and have a fraction . And so we
02:28	might talk about something like this , what if you
02:30	have a number , I'm gonna represent that number by
02:33	a letter B and I'm going to raise it to
02:35	the one half power . This is what we call
02:37	a rational exponents , because the exponent has a fraction
02:41	, the numerator of the fraction is one and the
02:43	denominator of that fraction is one half . So one
02:46	half there is in the exponent itself . Now we've
02:49	learned this in the past , but just in case
02:51	you haven't picked it up yet , when you have
02:53	a fractional exponent like this , let's say one half
02:55	power . The two here means it's gonna be a
02:58	square root . So these two things are interchangeable ,
03:03	right ? The B to the one half is exactly
03:06	the same thing as the square root of B .
03:08	Similarly , we have learned in the past that if
03:11	you have something like B to the one third power
03:14	, So now there's a three on the bottom instead
03:16	of a two . This is going to be uh
03:19	not the square root of B . It's going to
03:20	be the cube grew to be . These are identical
03:23	ideas . The one anything raised to the one third
03:25	power actually ends up becoming a cube root , right
03:29	? And then finally , just to kind of give
03:31	one more example , you might guess what would happen
03:33	if you raised B or anything to the 1/4 power
03:36	. What do you think it would be ? You
03:37	see the pattern here ? It's the fourth root of
03:40	being . Now . We have introduced these concepts in
03:42	the past when we talked about radicals , so it
03:45	shouldn't be completely foreign to you . But again ,
03:47	we're going into a little bit more detail . The
03:49	question I want to ask you is why is this
03:51	the case ? Why is it the case that a
03:53	fractional exponent is the same thing as a route .
03:56	You see the 1/4 power giving you 1/4 route .
04:00	One third power giving you a cube root and the
04:02	one half power . This is an implied too because
04:04	it's a square root . Why is that the case
04:07	? So let's take a second just to talk about
04:09	why that's the case . If this is actually true
04:12	, if it's true , then the following must also
04:16	be true . We can do anything we want .
04:18	If these are actually equivalent , this is an equation
04:20	. We can do whatever we want to both sides
04:22	. Right ? So let's take the B to the
04:24	one half power . Uh and let's raise it to
04:28	the second power . We can square the left hand
04:30	side of the equation , and then on the right
04:32	hand side of the equation will square it as well
04:34	. So you see , all we're doing is squaring
04:36	both sides of the equation . If this is actually
04:38	true , then this is a perfectly valid thing to
04:39	do . But you know that when you have an
04:41	exponent raised to an exponent , you just multiply the
04:44	exponents together . That's from basic uh exponents knowledge of
04:48	exponents that we've learned a long time ago , two
04:50	times one half is going to be 2/2 because the
04:54	two times the one and then the two times the
04:56	implied one . This is a 2/1 here . And
04:59	then what do you have on the right hand side
05:00	? We also know from our working with radicals that
05:02	if you have a square root and you square it
05:05	, that kind of undo each other . And so
05:07	you just get a B over here . But then
05:09	you can see that two to the power here ,
05:12	this is just the power of one . So this
05:13	is beating the one is equal to be . And
05:15	so then B is equal to be . So what
05:17	we have kind of shown is that a lot of
05:19	students say , well , why is this true ?
05:20	And here's kind of one proof of why it's true
05:23	, because if I square both sides of the equation
05:25	, I get exactly the same thing . And so
05:27	I get the identity that B is equal to be
05:30	. So that you need to sort of burn it
05:31	in your mind that any time you see a fraction
05:35	in an experiment , it is the same thing is
05:38	a radical . Their equivalent . There's no difference between
05:41	the two . It's like saying that I have ice
05:44	and I have water . They're both H20 . But
05:46	they're just slightly different representations of exactly the same thing
05:49	. When you see a radical . It's exactly the
05:52	same thing as an exponent . That is a fraction
05:54	. Okay , so that's kind of for this .
05:56	Now let's just fly through the other ones here because
05:58	you know why not ? We have a few minutes
06:00	. What if I take this guy this be to
06:02	the one third . What if I raise him to
06:05	the third power ? Then on the right hand side
06:08	if this is actually equivalent , I would have to
06:10	raise him to the third power . But I know
06:12	that this exponent will be multiplied by this exponent which
06:15	would be 3/3 . And I know that a cube
06:18	root cancels exactly with a cube . We all we
06:21	know from cubing things . We know that whenever we
06:24	raise to the power of the same base of the
06:26	of the of the radical there , they annihilate each
06:29	other and we're left with B . And so we
06:30	end up with B is equal to be because this
06:32	is the first power here . And you can imagine
06:34	the same exact thing would hold here . If I
06:36	raise this to the fourth power and raise this to
06:38	the fourth power , then I'll get B . Is
06:40	equal to be the same sort of thing as we
06:42	have done here . All right . So when you
06:45	see an exponent , that's one half it's a square
06:47	root . If you see an exponent , that's one
06:49	third . It's a cube root . If you see
06:51	an exponent that's 1/6 . It's 1/6 route . If
06:55	you see an exponent , that's 1/10 . Its 1/10
06:58	route . I mean you see the pattern , it's
06:59	not so hard to understand . Now . We need
07:02	to go beyond these basic exponents and talk about what
07:05	happens if I have something like what about something more
07:12	complicated ? I told you we're gonna go a little
07:13	deeper . What about B . Two the two thirds
07:16	? So this is different because in every example I
07:19	told you I said the one half power is a
07:21	square root . The one third power is the cuba
07:23	. At the 1/4 power is the fourth root and
07:25	so on . But this is not a one third
07:27	, it's two thirds . So that's different . Right
07:29	? So what do we have when we have something
07:31	like this ? Well I need you to think about
07:34	what the two thirds power really means . If I
07:38	have something like B to the two thirds power ,
07:41	how can I write this thing ? I can write
07:44	it as follows . I can write it is b
07:46	squared all raised to the one third power . How
07:50	do I know I can do that because remember exponents
07:52	whether they're fractional exponents like these or regular exponents ,
07:56	they all obey the same rules of exponents when you
07:58	have a power raised to a power like this you
08:01	just multiply the power . So we know that if
08:04	I multiply two times a third I'm gonna get two
08:06	thirds because the two times one and then he implied
08:09	one on the bottom here at times three I'll get
08:11	two thirds back . So this is exactly equivalent of
08:14	this . So I can kind of break these things
08:15	apart but then I know that the one third here
08:19	is a cube root . Right ? So then what
08:21	I'm saying here is that to be squared can then
08:23	be wrapped up underneath a cube root because the b
08:27	squared is underneath the cube root applies to the whole
08:30	thing . So what I'm saying is that b to
08:33	the two thirds power can be written like this but
08:35	it can also be written as the cube root of
08:37	B squared . However I can also write this another
08:42	way I can say that be to the two thirds
08:44	power can be written as B To the 1 3rd
08:49	raised to the second power . How do I know
08:52	I can do that because remember exponents , You just
08:54	multiply them . So two times a third gives you
08:57	two thirds . Just like two times a third gave
08:59	me the same two thirds here . All I've done
09:01	is reverse the order of what is inside and what
09:03	is outside . So this is exactly the same thing
09:07	as this which is exactly the same thing as this
09:09	which is exactly the same thing as this . Is
09:10	kind of like four different ways of writing the same
09:12	thing but if I write it like this then I
09:15	would take the cube root first , be just be
09:20	cube root of this and then whatever that is ,
09:22	that whole entire thing is squared like this . So
09:28	what I'm basically saying is that this representation and this
09:32	representation is the same . Thank literally what I'm saying
09:38	is there's no difference at all between this , this
09:41	this this this and this . You see where it
09:43	gets complicated , a lot of students look at that
09:45	and there and you try to memorize equations and formulas
09:48	and oh my gosh I'm gonna try to memorize it
09:50	. No don't memorize it . Just understand the fundamental
09:53	rules when you have an exponent raised to another exponent
09:56	you just multiply the exponents . So any time I
09:59	have a fractional power like two thirds I can write
10:03	it as the squaring coming first and then the cube
10:06	, the cube root part of it giving me this
10:08	. Or I can write it as the cube root
10:10	first and then square it . Which gives me something
10:12	like this . The reason I can do these in
10:14	any order is just because multiplying these together gives me
10:17	the same thing no matter which order I do it
10:20	. All right . So in your book you probably
10:23	will see something like this in general , whatever book
10:29	you're using will probably put something like this . If
10:31	I have be . This looks really confusing in my
10:34	opinion when you just read it in a book .
10:35	But now that we've done this , it won't be
10:36	hard at all if you have B to the P
10:40	over Q . Power . That looks crazy , doesn't
10:42	it ? What it's basically saying is that I can
10:45	write it like this be to the P . And
10:48	the cube root of that . Or I can write
10:51	it as b to the cube root of that .
10:55	Uh to the P . Power . Now this I
10:58	admit looks crazy . It looks it looks really cumbersome
11:01	and complicated . All it's saying is that if I
11:03	have a number or variable or whatever it is to
11:07	a fractional power , the numerator is P . And
11:09	the denominator is Q . All it's saying is that
11:12	I have to take the cute root of it because
11:14	that's on the bottom , that's a cube root or
11:17	fourth root or a square root whatever that is .
11:19	And I also have to raise B to the power
11:21	of P . Because that's on the top . But
11:23	what it's saying is it doesn't matter the order in
11:25	which I do that . I'm gonna get the same
11:26	exact thing . If I raise it to the power
11:29	first and then take the route , it's gonna be
11:32	the same thing is if I take the route first
11:34	and then raise it to a power , which is
11:36	exactly what I showed you here . In terms of
11:37	a number , example if you hide all of this
11:40	and I just give you this . It seems really
11:41	confusing . But you can see that with this ,
11:44	it didn't matter the order . And did it see
11:45	I squared it first and then I did the cube
11:47	root here , I give the cube root first and
11:50	then I squared it all . It's saying is that
11:52	when you have a fractional exponent we call a rational
11:54	exponents . It doesn't matter if you take the do
11:58	the square or the power operation and then the root
12:01	or the root and then the power . You're gonna
12:02	get the same answer if you grab a calculator and
12:05	actually do that both different ways , you will get
12:08	the same number because mathematically there they are the same
12:11	thing . So this is honestly the entire uh kind
12:16	of like the learning part of this lesson . That's
12:18	all I really want you to know now . What
12:19	we have to do is apply what we have kind
12:21	of learned here to some actual examples so we can
12:25	do that um Straight away . It's not gonna be
12:29	too bad . Let's start with something very very simple
12:32	. What if we have 81 to the one half
12:36	power ? How would we calculate that or simplify that
12:40	? Well the first thing we recognize is that the
12:42	one half power is just a square root . So
12:43	this is a square of 81 you all know that
12:45	nine times nine is 81 . Or you can write
12:48	this is nine times nine if you want to look
12:49	for a pair . So the answer is nine circle
12:52	, that is your answer . So 81 to the
12:54	one half power is nine . I encourage you grab
12:57	a calculator and actually take 81 and raise it To
13:01	the 05 power . That's one half right . And
13:04	you're going to find the answer is exactly nine .
13:06	That's what that's what comes out of it . All
13:08	right . Next problem . What if we have 49
13:14	raised to the negative one half power ? How do
13:17	we simplify that ? Well , we have two things
13:19	going on . We have a negative power and it's
13:21	also a fractional power . So , what this means
13:23	is since it's negative , we're gonna just drop this
13:25	guy downstairs and make it a positive one half power
13:29	. That's what happens with negative exponents . We drop
13:31	them down , make them positive . But this one
13:33	have power is just a square root . So this
13:36	becomes a squirt of 49 square root because it's a
13:38	two in the bottom of the fraction and seven times
13:42	seven is 40 9 . You all know that square
13:45	to 49 . And so you get 1/7 . This
13:50	is the answer . All right . So far ,
13:52	those are pretty elementary . Let's do something . Maybe
13:54	a teeny bit more challenging . What if you have
13:57	27 to the two thirds power ? This is the
14:00	first time when we have this exponent here . That's
14:04	a fraction . But it's it's not a simple one
14:06	, like one third or 1/4 . It's got the
14:07	two thirds . Now we learned just a second ago
14:10	that you can do this many different ways . I
14:12	can do the power first and then I can cubit
14:16	a cube root it because there's got to be a
14:18	cube root involved with the three in the bottom .
14:19	Or I could do the Q . Group first and
14:22	then do the power later . My advice is just
14:24	pick one . But I'm gonna do it both ways
14:26	to show you , you know what's happening here ?
14:29	Let's say the first thing we want to do is
14:31	square this this is a 27 and we're gonna raise
14:35	that to the power of to because there's a two
14:36	here . But then we're gonna wrap the whole thing
14:38	in the princes and raise the result of that to
14:40	the one third power . If I can write the
14:42	number three correctly , how do I know I can
14:45	do this ? Because if I multiply these exponents ,
14:47	I'm gonna get two thirds . That's exactly what I
14:49	started with . That's how you know that this is
14:51	legal to do . Now if you go into calculator
14:53	or grab a sheet of paper and square the number
14:55	27 . It comes out to be a really big
14:57	number 729 . That's big . But I have to
15:01	take the answer there and raise it to the one
15:03	third power , right ? And you all know that
15:06	the one third power is Just 729 cube root of
15:13	that . Now , how do I take the Cubans
15:14	? You have to take the Q . Bert of
15:15	this really big number . How do I do that
15:18	? Well , you got to grab a calculator or
15:19	something to figure out what multiplies together together to give
15:21	you 7 25 . When you play around with it
15:24	long enough , you're gonna realize that nine times 81
15:26	Works . And we all know that 81 is nine
15:29	times now . Now because it's a cube root ,
15:31	you're looking not for pairs , you're looking for triplets
15:34	and we found a triplet of nine . And so
15:37	the answer that we get is actually nine . So
15:39	if you go in your calculator and take 27 raise
15:42	it to the two thirds power . The exact if
15:44	you put the fraction is an exact two thirds and
15:46	raise it like this , you're going to get a
15:48	nine . If you take 27 square it and then
15:51	take the cube root of that , you're gonna get
15:52	nine uh there . But I mentioned that when we
15:56	did the squaring operation first it became cumbersome because this
15:59	number is big and then we have to find the
16:01	cube root of that really big number . So we
16:03	can do it another way we can do or the
16:06	following , we can say that 27 to the 2/3
16:09	power instead of squaring it first we can do the
16:12	other operation first , we can do that because of
16:15	the way the exponents work . We can raise it
16:17	to the one third power and then square the result
16:21	. How do we know ? We can do that
16:22	again because if I multiply these exponents together , I
16:24	get exactly what the problem statement was . So now
16:28	I have to take the cube root of 27 .
16:30	So let's just be explicit and write it down 27
16:33	cube root of this . And then the result of
16:37	that , I have to square it . Now 27
16:39	is a whole lot easier to take the cube root
16:41	of right ? Because you know that nine times three
16:43	is 27 and you know that nine is three times
16:45	three . These are things I have in my mind
16:47	, I don't know that nine times 81 is this
16:49	? I have to probably use a calculator for that
16:51	. But this is actually easy and I'm looking for
16:53	triplets and I found a triplet of threes . And
16:55	so what's going to happen Is this in the middle
16:58	is going to become a three And then I'm going
17:01	to be squaring that . And so I'm going to
17:03	get a nine and that's the answer . And notice
17:05	that this nine is exactly the same as this one
17:10	, because it doesn't matter the order in which you
17:12	do it . That's what I was trying to show
17:13	you here in terms of variables that you get the
17:16	same thing , no matter what you do . That's
17:18	what this is trying to tell you when you have
17:19	anything raised to a fractional exponent , you can either
17:22	raise it to a power and then take the route
17:25	, or you can take the route and then raise
17:26	it to the power . Same exact thing . That's
17:28	what we did , raise it to a power ,
17:30	then take a route . That's what we got .
17:33	Take the route , then raise it to the power
17:35	. That's what we got . Same exact answer .
17:38	All right . Usually it's going to be easier on
17:43	your on yourself if you have the choice just to
17:46	go ahead and take the route first notice that we
17:48	took the cube root first and that was easier .
17:51	We knew the cube root of 27 then we could
17:53	square and then we got the answer going . This
17:55	way we kind of required a calculator or a lot
17:58	of work on your separately to paper . So ,
18:01	if you have the choice of which one to do
18:04	first , usually you should go ahead and do the
18:06	radical first . Q . Bert square root for through
18:08	whatever you have and then do the other thing later
18:13	. All right . So let's keep on going .
18:15	Let's say we have the problem . 16 to the
18:19	3/4 power . Again , we can either cubit first
18:23	and then take the route , or we can take
18:25	the route for first and then we can cubit .
18:27	But we just learned that it's probably going to be
18:29	easier to write it like this . 16 to the
18:31	1/4 power will do the radical first and then we're
18:34	gonna cube the result . How do I know this
18:36	is legal ? Because if I multiply these exponents ,
18:39	I get three times one and one times four ,
18:41	I get 3/4 . So this is exactly equivalent to
18:43	this . And I know that the 4th root ,
18:46	I mean the 1/4 power is the 4th root of
18:49	16 . Of course , I still have two cubit
18:52	. And then I know , how do you take
18:54	the 43 to 16 while you just go down here
18:56	and say , well I know that 16 is four
18:57	times four , I know that four is two times
19:00	two and this is two times two . And since
19:02	it's 1/4 root , I'm not looking for pairs or
19:05	triplets , I'm looking for uh copies of four .
19:09	And so I found that um I have a two
19:13	that I can pull out of that radical , but
19:15	then that was just under here , I still need
19:17	to cubit two times 2 is four and then the
19:21	four times to again is eight , so two cube
19:23	is actually eight . That's the final answer . Now
19:26	, I'm choosing to do the radical first Because I
19:29	have choices . If I were to do the problem
19:31	again , of course I could take 16 and I
19:33	could cubit first , but that's gonna give you a
19:35	big number and then the big number is gonna you're
19:37	gonna have to take the fourth root of that under
19:39	a radical . Do a big factor tree . It's
19:41	going to be a little more work . So if
19:43	you can it's better to go ahead and do the
19:45	radical first . It saves you a little bit of
19:47	work . Mhm . All right , cranking right along
19:52	. We only have a few more of these .
19:53	What if we have -125 raised to the power of
19:59	-1 3rd . So notice that everything in this parentheses
20:04	is raised to this power and this power is itself
20:07	negative . So that means everything in here is under
20:10	kind of like the spell of the rat of the
20:13	experiment here . So because it's negative we're gonna drop
20:16	everything below And we're gonna make it negative 125 to
20:20	the positive 1 3rd . The negative comes with it
20:23	because it's in parentheses and the one third negative one
20:26	third . Excellent applies to the whole thing . So
20:28	you drop the whole thing down making a positive exponents
20:31	. All right . And then we know that one
20:33	third power means negative . 125 is a cube root
20:39	. Q route goes there because that's a one third
20:42	tower . And you might say , well wait a
20:44	minute . I thought we couldn't do radicals of negative
20:46	numbers . Well , it is true that if you
20:48	take the square root of a negative number , it's
20:50	the answer is not real . It's an imaginary number
20:53	. But this is not a square root . It's
20:55	a cube root . And the cube root of negative
20:57	numbers does exist . And let's see how you know
21:00	that is the case . So , if this were
21:02	just a positive 125 the way that you would write
21:05	it as you would say five times uh five times
21:10	55 times five times because this is 25 then 25
21:13	times 525 . But this isn't quite right because you
21:17	have a negative there . But if it's negative five
21:20	times negative five times negative five under this factor tree
21:25	, they all multiply together . Think of it this
21:27	way negative times negative positive but then positive times negative
21:30	makes it negative again . So this times this times
21:33	this actually does equal negative 1 25 and that's why
21:36	it does exist . And so what you're gonna get
21:38	is one over negative five or if you want to
21:41	be better about it right ? It is negative 1/5
21:43	with the negative sign is kind of sitting out in
21:45	the front there . Alright so again when you have
21:48	square roots of negative numbers you don't get real answers
21:50	. You get imaginary but cube roots perfectly fine to
21:53	take the cube root or fifth root or seven through
21:56	any odd power or any odd route . You can
21:59	take those of negative numbers . No problem . Okay
22:02	. Okay two more . What if we have four
22:08	To the negative ? 0.5 As an expert . Now
22:11	a lot of students will freeze up when they see
22:12	that because they see a decimal there and they're like
22:14	, what do I do ? Well this is an
22:16	exact No . is not an approximation , it's exactly
22:19	equal 2 -1/2 . And once you have it written
22:23	like this , you can drop it downstairs to be
22:25	positive one half . And then when you have it
22:29	down here you write it as the square root of
22:31	four because one half becomes a square root and then
22:34	the square to four is of course to so you
22:36	get the answer of one half . So if you
22:39	see a decimal that's exact like that . Just change
22:41	it into a fraction and kind of work through it
22:43	. Mhm . Now this following one will be our
22:45	last problem . What if we have negative eight To
22:50	the 2/3 power ? How do we simplify this guy
22:53	? Well there's a big gotcha in this problem and
22:55	you need to understand what it is . You see
22:57	here notice whenever we have this wrapped in parentheses that
23:00	we said the negative one third power applied to the
23:03	whole thing because including the negative side because it's wrapped
23:06	inside parentheses so we kind of had to bring the
23:09	negative along with it . However there is no parenthesis
23:11	here , so a lot of students will try to
23:14	apply this exponent to the negative that's here but that's
23:17	actually not right because there's no parentheses there . So
23:20	that negative is multiplied out in front but that exponent
23:23	does not apply to that because there is no parentheses
23:27	grouping them all together . So really a better way
23:30	to write this is you kind of put the negative
23:32	outside open the parentheses , bring the eight here And
23:35	make it 1/3 power . We're gonna do the route
23:37	first and then square it like this . So you
23:40	see the negative is not participating in the exponent because
23:43	it's just kind of sitting as a coefficient in the
23:45	front . The exponent is only applying to the eight
23:47	. So we kind of rapid and princes to kind
23:49	of force myself to recognize that and then this becomes
23:52	a cube root . So on the inside I'm going
23:55	to have a cube motivate and I have to square
23:59	the results now , what's the cube root of eight
24:01	? You all know that Eight is two times four
24:04	and four is two times two . And I'm looking
24:06	for triplets because it's a cube root . So there's
24:08	my triplet . And so what I'm going to have
24:12	Is the negative sign is still there . Inside the
24:15	Princess I just have a two because the Cuban 23
24:17	is too I have to square what is inside those
24:20	parentheses . The negative sign comes along the ride but
24:24	now I square the two and I get a for
24:27	the negative sign just stayed in front the whole time
24:29	. The answer to this guy is actually negative four
24:31	. If you go in a calculator or a computer
24:33	and you put a negative eight to the power of
24:35	two thirds you're gonna get a negative four . If
24:37	you get anything other than negative four then you put
24:39	it into the computer wrong because this negative does not
24:42	participate in the expo here . If I had wrapped
24:45	a princey around the negative and around the age so
24:48	that the whole thing was encapsulated to the two thirds
24:51	, then definitely it would it would have been different
24:53	answer . We're gonna have some problems like that in
24:55	a minute . But it would have been a different
24:57	order of operations . So basically it's not participating because
25:00	it's not wrapped like that . So the most important
25:03	thing for you to learn in this lesson or to
25:05	understand is that fractional exponents , which we call rational
25:09	exponents because rational number is a number that can be
25:12	written as a fraction . Basically the bottom denominator of
25:16	that fraction determines what route you're going to be taking
25:18	and the top of the fraction determines what power the
25:21	order of that you do the route and the power
25:24	can be whatever order that you want because of the
25:28	way exponents work . Because when you raise the power
25:30	to a power it doesn't matter the order in which
25:33	you do it . So usually though it's going to
25:36	be easier for you to take the route first before
25:38	raising the results of the power . As far as
25:40	like how much work you have to do . So
25:42	try to do that . If you can follow me
25:44	on to the next lesson , we have several lessons
25:45	here to get more practice with rational exponents , so
25:48	make sure you can solve all of these and follow
25:50	me on to the next lesson , we're gonna conquer
25:52	the rest right now .

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