Math Antics - Exponents and Square Roots - Free Educational videos for Students in K-12

Math Antics - Exponents and Square Roots - By Mathantics

Transcript


00:03	Uh huh . Hi , welcome to Math Antics .
00:08	In our last video called intro to exponents , we
00:11	learn that exponents also called indices are a special type
00:14	of math operation in this video we're going to expand
00:18	on what we know about exponents by learning about their
00:21	inverse operations . Which are called roots . That's kind
00:24	of a strange name for a math operation but it
00:26	will make more sense in just a minute first .
00:29	Let's review what we mean by inverse operations in the
00:32	video called what is arithmetic ? We learn that inverse
00:36	operations are pairs of math operations that undo each other
00:40	. For example , you can undo addition by doing
00:43	subtraction . So addition and subtraction are inverse operations .
00:48	Likewise , you can undo multiplication by doing division .
00:52	So multiplication and division are inverse operations . As I
00:57	mentioned , exponents have inverse operations . Also there are
01:00	operations that can undo them and those operations are called
01:04	roots to see how roots and exponents work together to
01:08	undo each other . Let's look at the simple exponent
01:11	for to the second power or for squared . Previously
01:15	we learned that this is the same as four times
01:17	four , which equals 16 . Doing this . Exponent
01:21	meant going from four squared to 16 . So now
01:25	if we want to undo that with the route operation
01:28	that involves starting out with 16 and then somehow getting
01:31	back to the four which is being raised to the
01:34	second power . And do you remember what that part
01:37	of the original exponent is called , yep , it's
01:39	called the base . So doing a root operation is
01:43	going to give us the base as our answer and
01:46	that helps us understand its name a little better .
01:49	The words base and root have a similar meaning ,
01:52	especially if you think of a tree , the route
01:55	is at the base of a tree and that can
01:57	help you remember how route operations work with the route
02:00	operation . You start with the answer of an exponent
02:04	and try to figure out what the base of that
02:06	original exponent is . Okay but how do we actually
02:09	do that ? How do we use a route operation
02:12	to go backwards and figure out the base of the
02:15	original exponent ? Well for starters we need to know
02:18	about a special math symbol that looks like this and
02:21	you guessed it , it's called the root sign .
02:23	Whoa dude that matt symbol looks totally radical . Dude
02:28	it's it's like that division thing . He only way
02:31	cooler . Ah yes that reminds me the root sign
02:35	is often referred to as the radical sign and mathematicians
02:39	use that term even before surfers did . And yes
02:43	it does look similar to the division sign so it's
02:46	really important not to get them confused . The route
02:49	or radical sign is different from the division sign because
02:53	instead of having a curved front it's front shape is
02:56	like a check mark . The number that you want
02:59	to take , the route of goes under the sign
03:01	like this . So when you see a number under
03:03	a root or radical sign like this , you know
03:06	you need to figure out the base of the original
03:08	exponent In this case you need to figure out what
03:11	number you can multiply together a certain number of times
03:14	to get 16 . Ah But there's the catch .
03:17	How many times the answer we get from taking the
03:20	route will depend on how many times that number would
03:23	be multiplied together , but that would depend on the
03:25	original exponent . So how do we know what that
03:28	number is ? Simple . The root symbol tells us
03:31	the root symbol actually includes the original exponent in it
03:35	. What ? You don't see it ? Oh that's
03:38	because I didn't try it in yet . And later
03:40	in this video you'll understand why . So let's put
03:43	a little too right here above the check mark part
03:46	of the root symbol And that two tells us that
03:49	we need to figure out what number or base could
03:52	be multiplied together two times in order to get 16
03:56	. And if you remember your multiplication table or if
03:59	you just look at our original example here , you'll
04:02	know that the answer to that is for . Now
04:05	, do you see how the route operation is ?
04:07	The inverse of the exponent operation ? When doing the
04:10	exponent ? We asked , what do we get if
04:13	we multiply four together two times and the answer was
04:16	16 . But when we did the route operation ,
04:19	we asked what number could we multiply together two times
04:23	to get 16 and the answer was four . Great
04:28	. Now that you understand how exponents and roots are
04:30	related , we're going to look closer at how rude
04:33	operations work to do that . We're going to change
04:35	our route problem slightly . Let's change the little to
04:39	into a little four . The first route was asking
04:42	us to figure out what number we can multiply together
04:45	two times to get 16 . But this new route
04:48	is asking us to figure out what number we can
04:51	multiply together four times to get 16 . That's a
04:55	bit trickier . Huh ? Can you think of a
04:56	number like that , yep . The answer is too
05:00	, because if you multiplied four twos together two times
05:03	two times two times two you get 16 . So
05:07	the second route of 16 is four , but the
05:10	4th root of 16 is too . Both those routes
05:14	were pretty easy to figure out right . But unfortunately
05:17	figuring out routes and math can be much harder .
05:20	For example , what if we had this problem instead
05:23	? Route three of 16 , That means we need
05:26	to figure out what number we can multiply together three
05:29	times to get 16 . Can you think of a
05:31	number like that ? No , I can't either .
05:34	And unfortunately it's not easy to calculate what that number
05:37	would be . Remember . Even though this looks a
05:40	little bit like the division symbol , this is not
05:43	just a vision . You can't just divide 16 x
05:46	3 to get the answer . Roots are not the
05:49	same as long division . So how do we calculate
05:52	a route like this ? Well , there are special
05:55	algorithms that you can use to calculate just about any
05:57	route , but they're kind of complicated . So we'll
06:00	save those for a future video instead , I'm going
06:03	to use a special route function on my calculator to
06:06	get the answer and on my calculator , the button
06:09	for that route function looks like this To use it
06:12	. I first entered the number that I want to
06:14	take , the route of which is 16 . Next
06:17	I hit the root function button and then I enter
06:20	three so it knows that I want the third root
06:23	of 16 . Last I hit the equal sign and
06:26	Voila the answer is 2.519842 . And the decimal digits
06:33	just keep on going forever Wow . See what I
06:36	mean about routes being hard to figure out . This
06:38	is a really complicated decimal number and you may even
06:41	wonder if it's the right answer . Well let's check
06:44	based on what we know about exponents and roots .
06:48	If we multiply this decimal number together three times we
06:51	should get 16 right . But to make it easier
06:54	to check , let's just round the number off to
06:56	two decimal places , let's make it 2.52 . If
07:01	we multiply a 2.52 together three times . In other
07:05	words , if we take 2.52 to the third power
07:08	will get 16.003 . Well that's almost right . It's
07:13	really close to 16 , isn't it ? The reason
07:16	it's not exactly 16 is that we rounded the number
07:19	off which made it less accurate but the more decimal
07:22	digits we use , the closer we'll get to 16
07:25	in math , the vast majority of roots are complicated
07:29	numbers like this and they're hard to figure out unless
07:32	you use a calculator or a special algorithm . That's
07:35	the bad news . But the good news is that
07:37	most of the time the routes will be asked to
07:39	do in your homework or on tests are the easy
07:42	ones , the ones that have nice hole number answers
07:45	and usually you'll only be asked to find the second
07:48	or third roots of numbers . Do you remember in
07:51	the last video we learn that two and three are
07:53	the most common exponents . So common . In fact
07:56	that they even had special names . Raising a number
07:59	to the second power was called squaring it and raising
08:03	a number to the third power was called cubing it
08:06	. Well , it's the same with roots since the
08:08	routes two and three are the most common , they
08:11	get special names . Also , the second route is
08:14	called the square root and the third route is called
08:17	the cube root . In fact , the square root
08:20	is so common that it's basically the default route and
08:24	its symbol even get special treatment . Do you remember
08:27	that when I first showed you the root symbol ,
08:29	I left out the index number that tells you what
08:32	route to find ? Well , whenever that number is
08:34	left out , you can just assume that it's too
08:37	. In other words , the root symbol with no
08:40	index number is always the square wound . So if
08:43	you want someone to find a different route like cubed
08:46	or fourth or fifth , then you need to include
08:49	that number . So they know which route to find
08:52	. And even those square roots are the most common
08:54	. They're not always easy to find . Most are
08:57	still going to be big . Long decimal numbers except
09:00	for the perfect squares . It's easier to find the
09:04	square roots of the perfect squares because their answers can
09:07	be found using the multiplication table on the multiplication table
09:11	. Have you ever noticed that all the answers to
09:13	problems where the same numbers being multiplied together are on
09:17	the diagonal of the table . In other words ,
09:20	two times two equals 43 times three equals 94 times
09:24	four equals 16 . 5 times five equals 25 .
09:27	Six times six equals 36 and so on . Well
09:30	, those numbers are called the perfect squares because they're
09:33	the answers you get when you square a whole number
09:37	. And that means if you take the square root
09:39	of a number along that diagonal , you get a
09:41	nice hole number . As your answer . The square
09:44	root of four is too The square root of nine
09:47	is 3 . The square root of 16 is four
09:51	. The square root of 25 is five and so
09:54	on . See what I mean . Those roots are
09:57	really common and they're also easy to figure out if
10:01	you know your multiplication facts . So if you're new
10:04	to exponents and routes , learning the perfect squares is
10:06	the place to start . Once you understand how those
10:09	exponents and roots work , you'll be ready to figure
10:11	out tougher problems . All right . So now ,
10:14	you know how exponents are related to roots , their
10:17	inverse operations and they undo one another . And you
10:21	also know that just like two and three are the
10:23	most common exponents . The square root and the cube
10:26	root are the most common roots . You also know
10:30	that finding roots is usually not very easy . That's
10:33	important to know . So you don't get discouraged if
10:35	you feel like it's hard to figure out what a
10:37	certain route is . You're not alone . We think
10:40	it's hard to and would normally just use a calculator
10:43	to find them . The good news is that some
10:46	roots are easy to find like the perfect squares .
10:49	So be sure to focus on learning them first and
10:51	remember to get good at math . You need to
10:53	actually practice what you learned from watching videos . So
10:56	be sure to do some exercise problems . And as
10:59	always , thanks for watching Math Antics and I'll see
11:01	you next time learn more at Math Antics dot com

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