What is an Exponent & Powers of 10? - [5] - Free Educational videos for Students in K-12

What is an Exponent & Powers of 10? - [5] - By Math and Science

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00:00	Hello . Welcome back . The title of this lesson
00:02	is called powers of 10 and understanding exponents . This
00:06	is part one . So we're doing two things that
00:08	are very important in this lesson . First we're gonna
00:11	start talking about the concept of an exponent . Which
00:14	sounds very complicated but I promise you it's not complicated
00:16	at all . And the second thing is we're talking
00:18	about powers of 10 . So when we use the
00:21	word powers of something it means the same thing is
00:24	using an exponent . So if you hear exponents or
00:27	if you hear something to a power then it means
00:30	they mean the same thing . So don't get confused
00:32	. The exponent and the power of 10 or power
00:35	of some number is the same exact thing . So
00:38	what I want to do is write down some exponents
00:40	on the board , make sure you understand what exponents
00:41	mean . And we're going to be talking about the
00:44	number 10 with exponents involved because it turns out that
00:48	powers of 10 have a ton of uses in math
00:52	and also in science and in engineering , we use
00:54	it all the time . You're just gonna have to
00:56	trust me . Powers of 10 is something you will
00:58	use forever . So let's talk about uh let's talk
01:03	about this idea of an exponent here . So if
01:05	I have the number 10 here , we're taking exponents
01:08	when we apply them to the number 10 . In
01:10	other words , powers of 10 . If we take
01:12	the number 10 and we raise it to an exponent
01:15	, which is a number four , what does this
01:17	actually mean ? So what it means is exponent ,
01:20	which is when you have the little number raised up
01:22	above . It just is a faster way of multiplying
01:25	something , you all know how to multiply things together
01:28	. But we do we do multiplication so much that
01:30	we have a short hand way of writing it down
01:33	, called an exponent . So when we have 10
01:35	raised to the power of four , what it actually
01:38	means if you write it all out is it means
01:41	10 times I'm gonna use a dot for the for
01:45	the multiplication . I'm not going to use the X
01:46	anymore because the X . We're going to be using
01:49	for other other things down the road . So it's
01:51	10 times 10 again , times 10 again , times
01:57	10 notice that I have four tens on the board
02:00	. That's because the exponents is a four . So
02:03	whenever you take the number 10 and you raise it
02:05	to a power of four , also called an exponent
02:07	of four . All you do is you take the
02:09	bottom number and you multiply it times itself and you
02:12	have that many of the of the exact same thing
02:15	multiplied times itself that many times in this case .
02:18	There's four tens here because the exponent is a four
02:21	. All right . Now , what do you think
02:24	is going to happen then ? If we have the
02:26	number 10 and we raise it to the power of
02:28	three ? Well , if this is 10 multiplied 1234
02:33	times 10 raised to the power of three is just
02:35	going to be 10 , multiplied by 10 , multiplied
02:39	by 10 . There's only 10 of the threes on
02:42	the board multiplied together because the exponent is a three
02:46	. So you can see the pattern here . What
02:47	do you think is gonna happen when we have a
02:48	10 and we raise it to the power of to
02:51	when you raise it to the power of to we
02:53	have another word we use for that . We say
02:55	it's 10 squared . When you see something is squared
02:59	, it just means the power or the exponent is
03:01	a two . That's all it is . When you
03:03	see something cubed . That's another word . That means
03:06	power of three . So you might hear something square
03:09	. That's the power of to something cubed is a
03:12	power of three . So what happens when we take
03:14	the number 10 and we square it ? In other
03:17	words we have the power of to then we just
03:19	take 10 and we multiply it by itself and there's
03:22	just two tins on the board . Now don't let
03:25	these dots confuse you . These dots . I remember
03:27	in the beginning always confused me but the dots just
03:29	mean multiplication . They just mean the same thing is
03:32	putting that X there . But we have to drop
03:34	the X . Because very soon we're going to be
03:36	using X is for other things in math and will
03:39	be very confusing to have X mean multiplication and also
03:42	for X to mean something else . So we're not
03:44	gonna do that anymore . We're going to drop that
03:47	now . Finally . What happens if you take 10
03:50	and we have a power of one ? What do
03:52	you think that's gonna be ? Well if this is
03:54	four tens multiply together and three tens multiply together and
03:57	to tens multiply together then what is 10 raised to
03:59	the power of one ? Well it's just 10 .
04:03	So you can see the pattern here . When you
04:04	have anything raised to the first power , it's just
04:07	the same number . When you have anything raised to
04:09	the second power it's two of them multiplied together cubed
04:13	. Which means power of three is three of them
04:14	multiply together . And when you raise to the power
04:16	of four it's four of them multiplied together . So
04:18	you see the pattern when you have something to the
04:20	fifth power , it'll be five of those things .
04:23	All multiplied together . When you have to the six
04:26	power to the seven power to the 10th . That
04:28	doesn't matter to the 250th power would just mean 250
04:32	of those things multiplied together times itself that many times
04:37	. That's what it means . So let's actually crank
04:39	through it now that we can kind of see this
04:41	10 raised to the power of one Is just 10
04:45	. So I'm actually gonna draw a little arrow here
04:48	and we're going to say that this is equal to
04:50	10 . Obviously it is equal to 10 because we
04:52	already wrote it here but this one is a little
04:54	bit different . What is 10 times 10 Now ,
04:57	if you were to get 10 and put 10 under
05:00	and multiply it the long way , zero times 00
05:03	times one drop zero . You do the long multiplication
05:06	, what you're going to get is 100 . That's
05:09	gonna be a 100 . When you take 10 times
05:12	10 , this is going to give you 100 .
05:13	We just figured this out . But if we multiply
05:15	it by 10 again , what you're going to get
05:17	when you do that is 1000 . And then of
05:20	course we already know that three tins multiplied . Give
05:23	us 1000 so that these three multiplied together are 1000
05:26	. If we multiply by 10 again , we're going
05:28	to get 10,000 . Now , I really want you
05:32	to study the pattern on the board because this is
05:34	why we actually study powers of 10 again . We
05:36	I say we use it in lots of science .
05:38	Chemistry , physics , math engineering . I mean over
05:41	and over again . There's lots of uses for this
05:43	. The reason why we uh we were doing this
05:46	here is I want to show you that when you
05:48	have Notice The pattern here , 10 raised to the
05:51	4th . Power Is just a one followed by four
05:55	Zeros . 10 to the third , power is just
05:57	a one with three zeros . To the second ,
06:00	power is just a one with two zeros . And
06:02	to the first power is just a one with 10
06:05	So you can very easily switch back and forth .
06:08	What we're saying is that the number 10,000 is equal
06:11	to 10 to the power of four . The number
06:14	1000 is 10 to the power of three . The
06:16	number 100 is equal to 10 to the power of
06:19	two . And the number 10 is just equal to
06:21	10 to the power of one . It's just the
06:23	number itself . And whether or not we write it
06:25	out like this or we write it like this is
06:28	exactly the same thing . Why do you think we
06:31	write things with powers and exponents ? It's because writing
06:34	big big numbers like this , you have to write
06:37	a lot of zeros And when we have , let's
06:39	say you're talking about the size of the universe or
06:41	the size of the solar system in kilometers , let's
06:44	say it might be one with a ton of zeros
06:48	at the end . You'll be writing zeros forever Endeavour
06:50	because it's so big , it's such a big number
06:52	. But when we can take this and write it
06:55	with an exponent we can shorten the number but it
06:58	means exactly the same thing and the pattern is whatever
07:01	the exponent is , it's just a one with that
07:03	many zeros with that many zeros with that many zeros
07:07	that follow . Okay , so now that we know
07:11	what powers of 10 really mean ? We want to
07:13	do a few problems . Okay , we're gonna be
07:15	doing some conversions . We want to convert for problem
07:17	number one , convert the number below to a power
07:20	of 10 . So I want to convert to a
07:22	power of 10 . So let's say you have the
07:24	number 1000 Now , you can look at the chart
07:27	of course , but I want to go through it
07:29	with you and see how can we uh how can
07:35	we write it as a power of 10 ? Okay
07:38	. Remember the pattern here . Is that when we
07:40	have 1000 ? It's just 10 to the power of
07:43	three because there's three zeros at the end , it's
07:45	the same as 10 times 10 times 10 . Right
07:48	? So what we can say is this is going
07:50	to be 10 to the power of what ? Three
07:53	? Why ? Because there's three zeros . So it's
07:56	a one followed by three zeros 10 to the power
07:59	of three . It's exactly what we already wrote on
08:00	the board here . And we already said that .
08:02	That's 10 times 10 times 10 . When you multiply
08:06	these together , you're going to get 1000 when we
08:08	write it in the power of 10 , it's 10
08:10	to the power of three . All right , let's
08:14	convert the following number uh , into a standard number
08:19	. What if I give you 10 to the power
08:21	of five ? And I ask , you don't write
08:23	it as a power of 10 ? Right as a
08:25	full number . Now , we don't have 10 to
08:27	the power of five here , but you can see
08:29	the pattern that all you do when you have 10
08:32	to the power of something is you put a number
08:34	with that many zeros at the end . But what
08:36	you really need to remember is that what this really
08:39	means is 10 times 10 times 10 times 10 times
08:45	10 , 12345 of them . That's what the exponent
08:48	means . And so if you're writing it as a
08:50	full blown number , it's one with five zeros after
08:54	. And so we have 12345 zeros after and you
08:58	put your comma in here . And so the answer
08:59	is 100 1000 . So 10 to the power of
09:02	five is 100,000 . And the reason that's 100,000 is
09:06	because this is 10 times 10 times 10 times 10
09:08	times then 10 times 10 is 100 100 times 10
09:11	is 1000 . 1000 times 10 is 10,000 and 10,000
09:17	times 10 again is 100,000 . That's the final answer
09:22	. Alright , problem number three , convert the number
09:25	below to a power of 10 . Well the number
09:28	is just 10 itself . So the answer that we
09:30	have is just going to be 10 to the power
09:33	of one . Remember when you have a number raised
09:36	to the number the exponent of one . It's just
09:39	You're not really multiplying it by anything . So the
09:41	answer is 10 and that's what we had on the
09:43	board here . 10 to the power of one is
09:45	just 10 because you're not multiplying it by anything .
09:47	And so we just write that there . You can
09:50	remember in the back of your mind that anything raised
09:53	to the one power is just the number itself because
09:56	it's really not multiplied times anything else . All right
10:00	, let's scoot along here to the next problem .
10:04	Let's take a look at what happens when we have
10:09	the number 10 raised to the power of to and
10:12	we want to write it as a standard number .
10:14	What does tend to the power of to really mean
10:18	? It means we have the number 10 and we
10:20	multiply it times itself and we only have two of
10:23	these tens because the power is two and 10 times
10:27	10 When you multiply that out is 100 so if
10:30	we were going to take 10 squared and write it
10:32	as a number , it would be 100 . Notice
10:34	that it just means there's two zeros after the one
10:37	. So you could skip this step . I'm showing
10:39	it to you . So you remember what the exponent
10:41	means . But really 10 to the power of to
10:43	you can just skip straight to writing one with two
10:45	zeros at the end . All right . Alright .
10:49	Next problem . Let's convert the following # two .
10:53	A power of 10 . We're gonna write down one
10:57	million . The number here is one million . How
11:00	do we write it as a power of 10 ?
11:01	Maybe it's one million miles to the nearest , you
11:04	know , uh star or to the nearest planet or
11:07	something like that . But we don't want to write
11:08	it as all the zeros here . How do we
11:11	do it ? Well , what we can do is
11:13	we can say we can write it as a power
11:14	of 10 . Well , put it as 10 and
11:16	the power needs to be the number of Zeros here
11:18	. 123456 10 to the power of six . Why
11:22	do I know that ? This is the case because
11:24	10 to the power six is 10 times 10 times
11:28	10 times 10 times 10 times 10 . There should
11:32	be 123456 of them multiplied together . So 10 times
11:37	10 is 100 times 10 is 1000 Times 10 is
11:40	10,000 times 10 is 100,000 times 10 more is a
11:45	million . And of course we can skip this by
11:47	just counting the zeros and writing it as we have
11:49	shown here . All right . Next problem . Let's
11:56	take this number 10 to the power of four and
11:59	write it in standard as a standard number . All
12:02	right . Well , 10 to the power for what
12:04	does it mean ? It means 10 multiplied by 10
12:07	. Multiplied by 10 . Multiplied by 10 . There's
12:10	just four of them because the power is four .
12:12	And so you can multiply them and you can show
12:14	what the answer is . Or you can just look
12:16	at this and say , well , it's going to
12:16	be a one followed by 1234 zeros . Put a
12:20	comma and the answer is 10,000 . That's actually why
12:23	we're learning powers of 10 . Because you can just
12:26	look at the power and you know what the number
12:28	is . When it's 10 raised to an exponent ,
12:31	you can immediately write it down . It's just gonna
12:33	be the number one with this mini zeroes following that
12:37	one . So that was 10,000 . All right .
12:40	So , we have the final few problems here .
12:45	Let's convert the following # two . A power of
12:47	10 . 100 comma 000 comma 000 This number ,
12:53	in words is 100 million . Right ? Because this
12:56	is the millions place . This is the 10 millions
12:59	place . And this is the 100 million place .
13:01	So , this is 100 million . How do we
13:03	write it as a power of 10 ? Well ,
13:05	we know that it's one followed by this many zero
13:08	. So we can write it is a 10 raised
13:10	to the power of 12345678 10 to the power of
13:14	eight . So , you don't have to write the
13:16	multiplication is out . If it's one followed by a
13:18	bunch of zeros , you can just simply say it's
13:20	10 raised to the power of how many zeros it
13:23	is . So in this case it's 10 to the
13:25	power of eight . Yes . Right . Next problem
13:30	. Let's take this exponent power of 10 and write
13:33	it as a full blown number 10 to the power
13:35	seven . What does that actually mean anyway ? Well
13:39	, it's 10 times 10 times 10 . That's three
13:42	of them . Here's four of them . Here's five
13:45	of them . Here's six of them in here .
13:47	712 , 10 . Multiplied by itself . You have
13:52	seven tins on the board and they're all multiplied by
13:55	themselves . So how do we write this number out
13:57	? Well , it's 10 raised to that power of
13:59	seven . So we can just simply say it's a
14:01	number one followed by this many zeros 1234567 Put a
14:08	comma after three . Put a comma after three .
14:10	And the answer you have is 10 million . That's
14:13	the answer . 10 million . All right . Only
14:16	two more problems . Let's say we want to convert
14:20	this to a power of 10 1000 comma 000 comma
14:25	000 We have This is actually a new place value
14:29	. We haven't talked about if this is ones tens
14:31	, hundreds thousands , 10 thousands 100 thousands . Here's
14:35	millions . Here's 10 millions . Here's 100 millions .
14:38	Here's a new place value called billions . You ever
14:41	heard of ? A billion dollars or a billion of
14:43	something ? A billion miles away . This is how
14:45	a billion looks . One with nine zeros at the
14:48	end . But if I want to write it as
14:49	a power of 10 , all I have to do
14:51	is say , well , it's gonna be 10 to
14:52	the power of nine . Why ? Because there's 1234567890
14:57	So we already know the pattern . 10 to the
14:59	power of nine is just going to be a one
15:01	with that many zeros , nine zeros at the end
15:03	. And the way that you I guess could check
15:06	yourself is you could say , well , this is
15:08	the same thing as 10 times 10 times 10 times
15:13	10 times 10 .

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