Math Antics - Intro To Exponents (aka Indices) - Free Educational videos for Students in K-12

Math Antics - Intro To Exponents (aka Indices) - By Mathantics

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00:03	Uh huh . Hi , welcome to Math Antics .
00:08	In this lesson we're going to learn about an important
00:10	math concept called exponents Sounds kind of weird . Huh
00:14	. Indeed it does . My good man . And
00:16	I'm afraid it's because you're pronouncing exponents incorrectly . The
00:19	proper pronunciation is indices . Indices . Oh yeah that's
00:26	right . In a lot of countries , exponents are
00:29	called indices . But the good news is that the
00:32	concept is the same no matter what term is used
00:34	. And since I'm usually an american I'll usually use
00:37	the term exponents in this video . But all occasionally
00:40	point out the other terminology to sound good to you
00:43	jolly good sir . Toodle pip . All right then
00:47	. But what are exponents or indices ? Well just
00:50	like the four main arithmetic operations that we use in
00:53	math . Exponents are a type of math operation .
00:56	They tell us what to do with numbers . Okay
00:59	. But what do they tell us to do ?
01:01	Exponents tell us to take a number and multiply it
01:04	by itself a certain number of times . In other
01:07	words , exponents are basically repeated multiplication . To help
01:11	you understand what I mean ? Let's review multiplication for
01:14	just a second . Do you remember how multiplication is
01:17	basically repeated addition ? For example , two times four
01:22	is a shorthand way of writing two plus two plus
01:24	two plus two . It's the same as 4/2 being
01:28	added together which equals a total of eight . Now
01:32	let's compare that to a similar exponents , yep that's
01:35	an exponent . And in a minute we'll talk about
01:38	why it looks like that and why the numbers are
01:39	different sizes . But for now you just need to
01:42	know that this exponent tells us to take the number
01:45	two and multiply it four times . In other words
01:48	, you take four twos and multiply them together two
01:52	times two times two times two , which would equal
01:55	16 . So multiplication shows repeated addition and exponents show
02:00	repeated multiplication . And we'll see more examples of how
02:04	exponents work in a minute . But first let's talk
02:06	a little bit about why exponents look the way they
02:09	do exponents involved two numbers . The first number is
02:12	the one that gets multiplied repeatedly a certain number of
02:15	times and it's written full size and it's called the
02:18	base . The second number tells us how many copies
02:21	of the base to multiply together . It's written much
02:23	smaller and up at the top of the line like
02:25	this and it's called the exponents or the index .
02:29	And when reading exponents , some interesting terminology is used
02:32	. This exponent would usually be read two to the
02:35	fourth power or just to to the fourth for short
02:38	. And this exponent would be red three to the
02:40	fifth power or just three to the fifth . And
02:43	this exponent would be red 10 to the sixth power
02:46	or just 10 to the sixth Get the idea oh
02:49	and you'll often hear people say that the number is
02:51	raised to a certain power , you know because it
02:54	sounds kind of cool , but it's just another way
02:56	of saying the same thing . Unfortunately that terminology has
03:00	created a little confusion . Traditionally you'd say you have
03:03	a base raised to a number called an exponent and
03:06	the answer you get by doing that is called the
03:08	power of that base . But when people started saying
03:11	things like two to the fourth power or two to
03:14	the power of four , it made it sound like
03:16	the exponents was also called the power . The result
03:19	is that nowadays you may hear people use the words
03:22	exponent and power to mean the same thing . It's
03:25	unfortunate that the terms have become so mixed up or
03:28	wonky , but as long as you understand that the
03:30	base is the number that gets multiplied repeatedly and the
03:33	exponent tells you how many of them to multiply .
03:36	You'll be in good shape . But I know what
03:39	some of you are thinking . Where is the operator
03:41	symbol ? If exponents are math operations , don't we
03:44	need a symbol that goes between them . Like there
03:47	is with multiplication and the other arithmetic operations . Well
03:51	yes and no . Most of the time we don't
03:53	need a special operation symbol because of the way the
03:56	numbers are written . Since the exponent is written smaller
03:59	and up at the top of the line , it
04:01	looks much different from a normal digit . So we
04:03	don't need to use a symbol in math whenever you
04:06	see a regular sized number with a smaller number up
04:08	into the right , you know it's an exponent and
04:12	once you get used to seeing exponents like this ,
04:14	it's easy to recognize them . The only real concern
04:17	is when writing exponents down on paper , when you're
04:20	trying to solve problems . If you're not careful or
04:23	have really messy handwriting , you might accidentally confuse an
04:26	exponent like two to the fifth with a two digit
04:29	number like 25 . And obviously that would be a
04:32	problem . But even if you're careful when writing exponents
04:36	, there are some times when you really do need
04:38	a special symbol , like when you type and expanded
04:41	into a computer in that case it's very common to
04:44	use the caret symbol as the exponent operator . The
04:48	caret symbol looks like this . So two to the
04:51	fifth power would be written as two carat five And
04:54	three to the fourth power would be written as three
04:57	carat four . And that notation is used all the
05:00	time in computer programming . Oh and one more thing
05:04	you should know before we move on Is that exponent
05:06	operations do not have the community of property . In
05:10	other words , you can't switch the order of the
05:12	numbers without getting a different answer . For example ,
05:15	two to the fifth . Power is two times two
05:18	times two times two times two , which equals 32
05:22	. But if we switch the numbers five to the
05:24	second power is five times five which equals 25 .
05:28	So exponents do not have the community of property .
05:32	Okay . Now that you know what exponents are or
05:36	indices if you prefer and you know how they're written
05:39	. It's time to talk about how they're used in
05:41	math . As I mentioned earlier . Exponents are a
05:44	way of doing repeated multiplication , 3 to the second
05:48	. Power is the same as three times 3 .
05:50	Three to the third . Power is the same as
05:53	three times three times three , three to the fourth
05:56	, power is the same as three times three times
05:58	three times three . Three to the fifth power is
06:01	the same as three times three times three times three
06:05	times three . See the pattern . So exponents can
06:08	save you a lot of writing when you need to
06:10	show repeated multiplication . But what about when you actually
06:14	need to do repeated multiplication ? Do exponents help then
06:18	? Well , yes and no . If you have
06:21	to actually figure out what three to the fifth power
06:23	is , you still need to multiply three together five
06:26	times and you can do that by hand . Or
06:28	you could use a calculator to help you . In
06:31	fact , if you have a multi function or scientific
06:33	calculator , it might have a button on it that
06:36	looks something like this . X . To the power
06:38	of why . Which makes calculating exponents really easy .
06:42	Let's try using that calculator function to figure out what
06:45	this experiment would be six to the fourth power which
06:48	is the same as 46 is multiplied together on most
06:52	calculators . This is how the exponent button works .
06:55	First you type in the number that's the base of
06:58	your exponent . So in this problem we type in
07:00	six and then you'd hit the exponent button which might
07:03	not do anything besides . Let the calculator know that
07:06	the next number you give it will be an exponent
07:08	. So next you enter the exponent which is four
07:11	in this case . Now all you have to do
07:13	is that the equal sign to get the answer which
07:15	is 1,296 , wow . As you can see knowing
07:20	how to do that with a calculator saved us a
07:22	lot of work And now we know that 46 is
07:25	multiplied together will be 1,296 . In the examples we've
07:31	seen so far , we've only had simple one digit
07:33	exponents like 23 or four . But exponents can be
07:37	any number you could have to to the 84th power
07:41	or 12 to the 516th power . In fact as
07:45	crazy as it sounds , you can even have exponents
07:48	that are decimals or negative numbers but we'll save advanced
07:51	exponents like that for future videos . In the last
07:55	part of this video , I want to focus on
07:56	the two most common exponents you'll encounter which are two
08:00	and three . That is you'll very often see a
08:02	number raised to the second power or to the third
08:05	power . In fact , those exponents are so common
08:09	that they even have special names . When a number
08:11	is raised to the second power we say it's being
08:14	squared . So five to the second power is also
08:17	referred to as five squared And 12 to the second
08:20	power would be 12 square . Now if you're new
08:23	to exponents that might sound kind of funny . But
08:26	can you think of a reason why the term squared
08:28	is used , yep . That's how you would calculate
08:32	the area of a square shape . Since squares have
08:35	sides that are all the same length . If you
08:37	multiply that length together twice like four times four or
08:40	five times five or six times six it gives you
08:44	the area of that square . And it's a similar
08:47	story for numbers that are raised to the 3rd power
08:49	. In that case you would say that the number
08:51	is being cubed . So five to the third power
08:54	is also five cubed and eight to the third power
08:58	would be eight cubed . Since cubes are three dimensional
09:01	objects with sides that are all the same length .
09:04	If you multiply that length together three times you would
09:07	get the volume of the cube . So squaring the
09:10	number gets its name from squares and Cuban and number
09:14	gets its name from cubes . And that helps explain
09:17	why the exponents two and 3 are really common in
09:20	math because we interact with two and three dimensional objects
09:24	all the time in the real world . There are
09:26	lots of practical applications . Okay So now you know
09:30	that exponents are a way of showing repeated multiplication and
09:34	you also know what the two numbers in an exponent
09:36	means the base is the number that will get repeatedly
09:39	multiplied together . And the exponents or index tells us
09:43	how many times to repeat it in the next video
09:46	we'll learn more about exponents and their inverse operations called
09:50	roots learning Math takes a lot of practice . So
09:53	be sure to practice what you've learned in this video
09:55	. Thanks for watching Math Antics and I'll see you
09:57	next time learn more at math Antics dot com

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