Algebra Basics: Laws Of Exponents - Math Antics - Free Educational videos for Students in K-12

Algebra Basics: Laws Of Exponents - Math Antics - By Mathantics

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00:03	Do you know why I pulled you over today somewhat
00:06	? That's right . A law , this law to
00:08	be exact laws of exponents ? I've never even heard
00:12	of these . How was I supposed to know ?
00:14	You must have never heard of math antics . There's
00:16	this really cool math video series . They have all
00:19	sorts of basic math videos and the host is really
00:22	funny . Is this some sort of add , can
00:25	I skip this ? No , I think you'll really
00:26	like it , check it out . Mhm . Mm
00:32	Hi , this is rob . Welcome to Math Antics
00:35	in this video . We're going to learn about the
00:37	laws of exponents . If you look up the laws
00:39	of exponents online or in a math book , you'll
00:42	probably see a long list of equations that looks something
00:44	like this , wow , That's kind of overwhelming when
00:48	you see them all at once . But don't worry
00:50	, we'll take them one step at a time and
00:52	you'll see that they're not that complicated after all .
00:54	But before we get going , if you're not confident
00:57	with the basics of exponents , I highly recommend watching
01:00	our previous videos about them . Before moving on in
01:02	this video . Okay , let's start with just the
01:05	first two laws on our list . Which should look
01:07	familiar . If you watched our video called exponents in
01:10	algebra , they're just the two rules we learned in
01:13	that video . You know , rules laws , same
01:16	difference . So you probably already know them . They
01:19	simply tell us that anything raised to the first power
01:21	is itself and anything raised to the zero with power
01:24	is just one . And because we already know about
01:27	exponents that are higher energy values like X . To
01:30	the second X to the third . That means we've
01:32	pretty much got things covered , right ? Ah Not
01:35	so fast . Don't forget that integers can have negative
01:38	values too . For example what if we had the
01:40	expression X . To the power of negative one or
01:44	negative two ? Or negative three ? We know that
01:47	exponents are a way of doing repeated multiplication . But
01:50	how in the world could you multiply something together ?
01:52	A negative number of times ? Well you can't fortunately
01:57	. The next law on our list tells us how
01:58	to interpret a negative exponent . That law says X
02:01	to the negative and power equals one divided by X
02:05	to the power . And if you think about it
02:07	, that kind of makes sense . A negative number
02:10	is the inverse of its positive counterpart . And division
02:13	is the inverse operation of multiplication . Right ? So
02:16	a negative exponents is basically repeated , division X to
02:20	the negative one would be one divided by X .
02:23	X . To the negative two would be one divided
02:25	by X divided by X . X . To the
02:28	negative three would be one divided by X divided by
02:30	X divided by X . And so on . Seeing
02:33	it like this makes the pattern clear . But mathematicians
02:36	prefer to express negative exponents infraction form where one is
02:40	divided by the same number of X is multiplied together
02:43	. But since those multiplied exes are all on the
02:45	bottom of the fraction you're actually dividing by all of
02:48	them . Here's an example that will help you see
02:51	that that's true To to the negative 3rd power .
02:54	Let's first try that as a repeated division problem .
02:56	Like our pattern chose us . We always start with
02:59	a one so we would have one divided by two
03:02	divided by two divided by two . If we do
03:04	those operations from left to right using a calculator we
03:07	get 0.125 as the answer . Now let's write an
03:11	infraction for him . Like our law of exponents tells
03:13	us we can to to the negative third power would
03:16	be the same as one divided by two to the
03:19	third power . And that's the same as 1/2 times
03:22	two times two . Which simplifies to 1/8 and 1/8
03:26	simplifies to 0.125 See whether you write it as a
03:30	pattern of repeated division or infraction form . Like our
03:33	inverse law shows you get the same answer . And
03:36	now you know how to handle any expression with a
03:38	negative exponents . It's just one over the same expression
03:42	with a positive exponent X . To the negative one
03:45	is one over X . To the positive one .
03:47	Or just one over X . X . To the
03:49	negative two is one over X . To the positive
03:52	too X to the -3 is one over X to
03:55	the positive three and so on . All right .
03:58	Three laws down five more to go . And these
04:01	next five show us how we can do various math
04:03	operations involving exponents . In fact , the next law
04:07	tells us how we can take a number raised to
04:09	a power and then raise that to a power .
04:12	As you can see . It shows an expression X
04:14	to the power of them grouped inside parentheses . And
04:17	then that whole group is being raised to the end
04:19	power . It's a nesting situation . Kind of like
04:22	those Russian nesting dolls . So what if someone asked
04:25	you to simplify the expression X squared cubed ? Which
04:29	means the entire X squared term is raised to the
04:31	third power ? Well , our law tells us that
04:34	we can simplify that by multiplying the exponents together .
04:37	See how it equals X . To the power of
04:39	Mn . Which means M . Times in that means
04:43	X squared raised to the third power would be the
04:46	same as X to the power of two times three
04:48	, which is six . I want to see why
04:50	that's true . Well , think about what it would
04:53	mean to raise X squared to the third power .
04:56	It would mean multiplying three X squared terms together like
04:59	this . And each one of those X squared terms
05:02	simplifies to X times X . Right ? So we
05:05	end up with six X is being multiplied together which
05:08	is just X to the sixth power . See our
05:11	law works great . If you have a number raised
05:13	to a power and that's all raised to another power
05:16	. You can just multiply the two exponents together to
05:18	simplify it . And it works for negative exponents to
05:21	like what if we had X squared raised to the
05:24	negative third power . Well , our law tells us
05:26	that that's the same as X to the power of
05:29	two times negative three Which is X to the -6
05:33	. To see if that's true . We'll need to
05:35	use the law . We just learned about negative exponents
05:38	and rewrite this as one over X squared to the
05:41	positive third power . That simplifies to one over X
05:44	squared times X squared times X squared . Which in
05:47	turn simplifies to 1/6 X . Is being multiplied together
05:51	. And that all checks out because one over X
05:54	to the sixth would be the same as X .
05:56	To the negative sixth power . Pretty cool . Huh
05:59	? Okay . We're halfway through our list of laws
06:02	and we're going to look at the next to as
06:03	a set because they tell us how we can multiply
06:06	and divide expressions that have the same base . And
06:09	that's important because we can simplify them to have a
06:11	single exponents if the bases were different . The first
06:14	law says that if we have the base X with
06:17	exponents m being multiplied by the same base X With
06:21	exponents in , we can combine them simply by adding
06:24	the exponents together . And the second law says if
06:28	we have the base X . With exponents M .
06:30	Being divided by the same base X with exponent N
06:34	. We can combine them simply by subtracting the exponents
06:38	. Let's see some examples of each like this 12
06:41	to the third times two to the fourth . Does
06:44	that fit the pattern of our first law ? Yep
06:46	. The base of both expressions is the same but
06:49	they happen to have different exponents . The law would
06:52	still work if the exponents were the same but they
06:54	don't have to be just the bases have to be
06:56	the same . Our law tells us that this would
06:59	equal two to the power of three plus four or
07:03	two to the seventh power . But does it well
07:05	let's break it down and see two to the third
07:08	is two times two times two . And that's being
07:10	multiplied by two to the fourth which is two times
07:13	two times two times two . That's a lot of
07:15	two is being multiplied together . 72 is to be
07:18	exact ha ha . So that is what you get
07:20	by just adding the exponents together . Since three plus
07:23	four equals seven . And this law makes total sense
07:26	. If you think about what an exponent really means
07:29	. The exponent is telling you to do repeated multiplication
07:32	of the base . Right ? So this first part
07:35	is telling you to multiply three twos together . And
07:37	the second part is telling you to multiply four twos
07:40	together . So that's why you can add the exponents
07:43	together if the base is the same or think about
07:46	it like this . If we had 10 X .
07:49	Is all being multiplied together . We could form different
07:52	groups of them and combine them using exponents . Like
07:54	we could combine the first four exes into X to
07:57	the fourth and combine the remaining six . X .
07:59	Is into X to the sixth . And of course
08:02	those expressions would be multiplied together since all of the
08:05	exes were being multiplied . But you'd probably never want
08:08	to do that , would you ? I mean why
08:10	not just combine all 10 X is into the expression
08:12	X to the 10th . Ah But there you see
08:15	that our law holds true . X to the fourth
08:17	times X . To the sixth . Would equal X
08:20	to the power of four plus six or X to
08:22	the 10th . Okay now let's move on and see
08:25	some examples of the second law in this set which
08:27	tells us how to divide expressions with the same base
08:30	. Suppose we have the expression five to the third
08:33	power divided by five to the second power . Our
08:36	law says that we can simplify this by subtracting the
08:39	exponents . Specifically we take the exponent on the top
08:42	and subtract the exponents on the bottom from it .
08:45	If we do that , the simplified version would be
08:47	five to the power of 3 -2 or five to
08:50	the first power . But is that right to see
08:54	let's write the expression out in expanded form on the
08:57	top of our division problem . We have five to
08:59	the third which is five times five times five .
09:01	And on the bottom we have five to the second
09:04	which is five times five . Does this look like
09:06	something you've seen while simplifying fractions , yep . Since
09:09	all the bases are the same , they form pairs
09:12	of common factors on the top and bottom . That
09:14	can be cancelled out . This 5/5 cancels and this
09:17	5/5 cancels . If you don't know why that works
09:20	. Be sure to watch our video about simplifying fractions
09:23	and what do we end up with ? Well all
09:25	the factors on the bottom cancelled out . Which leaves
09:27	one since there is always a factor of one and
09:30	there's only 15 left on the top . So our
09:32	expression simplified to 5/1 or just five . Following our
09:37	law , we got a simplified version of five to
09:39	the first power which is also just five . So
09:41	it really did work . But to make sure you've
09:44	really got it , Let's try using this law again
09:46	with the expression X . to the fourth power over
09:48	X to the 6th power . This one's interesting .
09:52	Our law says that we can simplify it by subtracting
09:54	the bottom X moment from the top right . But
09:57	in this case that will give us a negative exponents
09:59	. Because the bottom exponent is bigger than the top
10:02	4 -6 would be negative too . So according to
10:05	our law , this expression should be equal to X
10:08	to the negative too . Let's try writing it out
10:11	in expanded form to see if that's true . On
10:13	the top X to the fourth would be the same
10:16	as four X is multiplied together and on the bottom
10:18	X to the sixth would be the same as six
10:20	. X is multiplied together . Once again , we
10:23	see that there are pairs of common factors that we
10:25	can cancel or pairs to be precise . And when
10:28	we cancel them all , we're left with a one
10:30	on the top because there's always a factor of one
10:33	and only two Xs multiplied together on the bottom .
10:36	If we recombine these two exes , we get one
10:38	over X squared . And if you remember the law
10:40	we learnt earlier about negative exponents . You'll see that
10:43	one over X squared is exactly the same as X
10:46	. To the power of negative two . So these
10:49	laws really do work okay . It's finally time to
10:52	look at the last two laws on our list and
10:54	fortunately there pretty easy ones , so we're not going
10:57	to spend too much time on them . These laws
10:59	look kind of similar to the last pair , just
11:01	like before . The first one involves multiplication and the
11:04	second one involves division . But notice that in these
11:07	laws the bases are different , but the exponents are
11:09	the same . That's the exact opposite of the situation
11:12	with the last pair of laws . It turns out
11:14	that these laws aren't about simplifying exponents there about how
11:17	you can distribute or un distribute a common exponent to
11:20	different bases . The first law shows this group X
11:24	times Y . That's being raised to the power of
11:26	em . And it says that you can rewrite it
11:28	as X . To the M times why to the
11:30	M . In other words you can distribute the exponent
11:33	to each factor in the group . And the second
11:36	law shows the group X divided by Y . That's
11:38	being raised to the power of in . And it
11:40	says that you can rewrite it as X to the
11:42	in divided by Y . To the in in other
11:45	words you can distribute the exponents to each part of
11:47	the fraction . Of course these laws would work in
11:50	reverse too and you could undistributed exponents if they're the
11:53	same . For example , if you're given the expression
11:57	X squared times y squared , you could rewrite that
12:00	as the quantity X times Y squared . And if
12:03	you have the expression X squared divided by Y squared
12:06	, you could rewrite that as the quantity X over
12:08	Y squared . There are two expressions that will help
12:12	you see why you can distribute or un distribute exponents
12:15	like our laws show in the first expression we have
12:18	the quantity X times Y squared and that's the same
12:21	as X times Y times X times Y . And
12:24	the community of property says that we can rearrange those
12:27	factors like this , X times X times Y times
12:30	Y . But look , we can simplify that into
12:33	X squared times Y squared . So that checks out
12:36	in the second expression we have the quantity X over
12:39	Y squared . That's the same as X over Y
12:42	times X over Y . To multiply these fractions .
12:45	We just multiply the tops and multiply the bottoms .
12:48	Which gives us X times X over Y times Y
12:51	. And that simplifies to X squared over white square
12:54	. So that one checks out too . All right
12:57	. So now , you know about the so called
12:59	laws of exponents and there's a good chance that you'll
13:01	see them explained in slightly different forms or different orders
13:05	or even using different terminology in other math videos or
13:08	books . But the basic ideas will be the same
13:11	. Some people like to try to memorize this list
13:14	of laws and you can do that if you want
13:15	to , but it's an even better idea to focus
13:18	on knowing how exponents really work , because if you
13:21	truly understand that , you can actually figure out a
13:23	lot of these laws for yourself and what's the best
13:26	way to understand how exponents really work , yep ,
13:29	you gotta practice . So be sure to do some
13:31	problems with exponents on your own . As always .
13:34	Thanks for watching Math Antics and I'll see you next
13:36	time learn more at Math Antics dot com . So
13:41	what do you think ? I thought you said the
13:43	guy was going to be funny .

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