Square Roots | Intro, Perfect Squares, and Simplifying | Math with Mr. J - Free Educational videos for Students in K-12

Square Roots | Intro, Perfect Squares, and Simplifying | Math with Mr. J - By Math with Mr. J

Transcript


00:0-1	Welcome to Math with mr J . In this video
00:05	, I'm going to give a general overview of square
00:08	roots . So we're going to start with what our
00:11	square roots then take a look at finding the square
00:14	root of a perfect square . And then lastly simplifying
00:18	square roots . So let's start with what our square
00:22	roots . Now , when we're looking for the square
00:25	root of a given number , we need to think
00:27	about what number multiplied by itself gives us that given
00:33	number . That may not make any sense now ,
00:36	but after our examples , you'll see exactly what I
00:40	mean . So let's jump in the number one Where
00:42	we have a three in the nine . Before we
00:46	talk about square roots , let's talk about squaring a
00:49	number , squaring a number means we have an exponent
00:53	of two . That means we multiply the number by
00:57	itself . For example , three squared means three times
01:02	three . So let's do this . Three squared means
01:09	three times three , which gives us nine . Right
01:12	? Three squared equals nine . Now , let's start
01:16	with that nine and do the opposite or inverse of
01:19	squaring a number . And that's going to be taking
01:23	the square root . So let's start with nine and
01:26	take the square root . So that's going to give
01:29	us well the square root of nine . Let's think
01:33	about what number multiplied by itself , will equal the
01:38	number under the square root symbol . This is the
01:42	square root symbol there , also known as the root
01:45	symbol or radical symbol . Well , we know three
01:49	times three equals nine . So the square root of
01:53	nine is three . Think about it . We know
01:56	three times three equals nine . So a number times
02:03	itself equals the number under the square root symbol .
02:08	So the square root of nine equals 3 . Let's
02:12	move on to number two and try another one .
02:16	So five squared five times 5 equals 25 . Let's
02:23	start with 25 and take the square root . So
02:27	the square root of 25 What number times itself equals
02:35	25 ? Well , we know five times five equals
02:39	25 . So the square root of 25 equals five
02:45	. Five times five equals 25 . A number times
02:51	itself gives us the number under the square root symbol
02:55	. So again , the square root of 25 equals
02:59	five . Let's move on to numbers three and four
03:02	. And we're just going to take the square root
03:05	of these numbers . So for number three , we
03:07	have the square root of four . Well , we
03:10	know two times two equals 42 times two equals four
03:17	. A number times itself equals the number under the
03:22	square root symbol . So the square root of four
03:26	Equals two . That's our final answer . And lastly
03:31	number four , we have the square root of 36
03:35	. Well , we know six times six equals 36
03:41	. A number times itself Equals the number under the
03:45	square root symbol . So the square root of 36
03:49	equals six . Now that we know a little bit
03:53	more about square roots , we're going to take a
03:55	look at more examples of finding square roots of perfect
03:59	squares . So let's jump into it . Let's start
04:03	with number one . Where we have the square root
04:06	of nine . So we need to think about what
04:09	number multiplied by itself will equal nine . So we
04:14	know that three times three equals nine . Right ?
04:19	Three times three Equals nine . So that's what I
04:23	mean by a number three multiplied by itself gives us
04:29	the number under the square root symbol . So the
04:32	square root of nine equals 3 . And that's our
04:37	answer . Let's move on to number two . Where
04:41	we have the square root of 16 . So we
04:45	need to think again . What number multiplied by itself
04:49	equals 16 . We know four times 4 equals 16
04:54	. Four times four equals 16 . So a number
05:00	multiplied by itself gives us the number under the square
05:05	root symbol . So the square root of 16 equals
05:09	four . And that's our final answer . Let's move
05:13	on to numbers three and four where we have four
05:17	square roots for each Three of the four will have
05:21	square roots of perfect squares , meaning they have a
05:25	whole number answer and one will not work out so
05:28	nicely . Let's solve these and find which ones are
05:32	the perfect squares and which one is not . So
05:36	we'll start with number three and first we have the
05:39	square root of 81 . So we need to think
05:42	any numbers that multiply by themselves to get 81 ,
05:47	Well , nine times nine . So 81 is a
05:51	perfect square . So the square root of 81 equals
05:55	nine . So we get a whole number answer there
05:58	and that is a perfect square . So I will
06:01	put a check there . Moving on to the Square
06:04	Root of 26 . So we need to think any
06:08	numbers that multiply by itself to equal 26 . Well
06:13	, we know five times five equals 25 . That's
06:16	close , but not quite . And then six times
06:19	six equals 36 . So the square root of 26
06:25	Is going to be somewhere between five and 6 .
06:28	It's not a perfect square and it's not going to
06:31	give us a whole number answer . So that's the
06:34	square root . That is not a perfect square within
06:38	these four . Let's do the other two which are
06:42	perfect squares . So the square root of 49 ,
06:45	7 times seven equals 49 . So the square root
06:50	of 49 equals seven . And this is a perfect
06:57	square . Lastly , we have the square root of
07:00	100 and we know 10 times 10 equals 100 .
07:05	So the square root of 100 equals 10 and this
07:10	is a perfect square . Lastly # four , we
07:15	will start with the square root of four . So
07:19	is there anything that we multiply by itself to equal
07:21	for ? Yes , two times two equals four .
07:24	So the square root of four Equals two . And
07:29	this is a perfect square . Now we have the
07:33	square root of 144 , well 12 times 12 equals
07:39	144 . So this is a perfect square . The
07:43	square root of 144 Equals 12 and it is a
07:48	perfect square . Now we have the square root of
07:52	64 , eight times 8 equals 64 . So this
07:57	is a perfect square . The square root of 64
08:03	equals eight . And then lastly we have the square
08:07	root of 74 . This is not a perfect square
08:11	. So we're not going to get a whole number
08:12	answer here . The square root of 74 is going
08:16	to be somewhere between eight and nine because eight squared
08:21	equals 64 then nine squared equals 81 . So again
08:27	, it's going to be somewhere between eight and nine
08:30	. This is not a perfect square . So we're
08:33	not going to get a whole number answer . So
08:36	there you have it . There are some examples of
08:39	how to find the square root of a perfect square
08:42	. Now that we have a better understanding of square
08:45	roots and working with perfect squares , we're going to
08:48	move on to simplifying square roots . So let's jump
08:51	into that . In this section , we're going to
08:54	take a look at simplifying square roots and we have
08:57	four examples that we're going to go through together in
09:00	order to get this down . So let's jump in
09:03	the number one where we have the square root of
09:06	20 . Now 20 is not a perfect square .
09:10	So we're not going to get a nice and clean
09:12	cut hole number answer . So we need to simplify
09:16	, we can do this by looking for factors of
09:19	20 that are perfect squares and then find their square
09:23	roots . For example , we know that four times
09:26	five equals 24 5 are factors of 20 . They
09:31	go into 20 so to speak , four is a
09:34	perfect square . So let's do this in order to
09:38	simplify the square root of 20 equals the square root
09:44	of four times 5 . Now the multiplication or product
09:50	property of square roots lets us split this , meaning
09:55	we can do this the square root of four times
09:59	the square root of five . That's equivalent to the
10:03	square root of 20 . It's equal . We're not
10:05	changing the value of the problem at all . Now
10:09	we can take the square root of our perfect square
10:13	four and we end up with , Well the square
10:16	root of four is too bring down our square root
10:21	of five because we cannot simplify that any further .
10:25	So our simplified answer is two times the square root
10:29	of five or two square root five . Let's try
10:34	another one and move on to number two where we
10:36	have the square root of 32 . So we need
10:39	to think are there any factors of 32 that are
10:42	perfect square ? So any numbers that can go into
10:46	32 , so to speak , that are perfect squares
10:50	. If so , we can simplify if not .
10:53	It's already in simplest form . Well , we know
10:56	that 16 times two equals 32 and 16 is a
11:00	perfect square . So we can simplify the square root
11:04	of 32 Equals The Square Root of 16 Times two
11:12	. Let's split this . So we have the square
11:14	root of 16 Times The Square Root of two .
11:20	16 is a perfect square , so the square root
11:23	of 16 is four , so we end up with
11:26	four times the square root of two or four square
11:32	root two . Or you can even say for route
11:34	to now two cannot be simplified any further . So
11:38	this is our final answer As far as simplifying the
11:43	square root of 32 . Now I do want to
11:46	mention we can take another path to get to that
11:49	same answer for number two . I'm going to try
11:52	to squeeze it in to the left here . We
11:55	know that four times eight also equals 32 4 is
11:59	a perfect square . So this is the other path
12:02	we can take . And I'm going to start by
12:04	splitting these just to make sure I have enough room
12:07	. So we have the square root of four times
12:12	the square root of eight . The square root of
12:15	four is too Let's bring down our square root of
12:20	eight . Now we can continue to simplify . We're
12:24	not done yet because we have a perfect square within
12:27	eight as far as factors go , because four times
12:31	two equals eight and four is a perfect square .
12:34	So again we can continue to simplify . So we
12:38	have two times the square root of four Times The
12:45	Square Root of two . So the square root of
12:48	four again is too , so we end up with
12:51	two times two times the square root of two .
12:55	Now the square root of two is simplified So we
12:59	can't break that down any further but we can multiply
13:03	our two times two , So two times two gives
13:06	us four And then we have the square root of
13:10	two . So same exact answer , but a different
13:14	path . And that's okay . Both of those paths
13:17	are correct . So keep that in mind . There
13:19	may be different paths as far as simplifying as square
13:22	root . Just always remember to check if you can
13:25	simplify further . Let's move on to number three and
13:29	try another one where we have the square root of
13:32	45 . So think any factors that are perfect squares
13:37	? Yes , nine and five or factors of 45
13:40	9 times five equals 45 9 is a perfect square
13:45	. So the square root of 45 equals the square
13:49	root Of nine times 5 . Let's split this .
13:53	So square root of nine times the square root of
13:59	five . So the square root of nine is 3
14:04	. Bring down our square root of five . That
14:07	cannot be simplified any further . So we have three
14:11	times the square root of five or three square root
14:14	five or three . Route five . Lastly number four
14:19	, we have the square root of 75 . So
14:22	any factors that are perfect squares , well , 25
14:27	3 are factors of 75 25 times three equals 75
14:32	25 is a perfect square . So the square root
14:36	of 75 equals the square root of 25 times three
14:45	. Let's split . So we get The square root
14:48	of 25 Times The Square Root of three . Now
14:53	the square root of 25 is five , So we
14:56	have five and then the square root of three is
15:01	in simplest form , so we can't break that down
15:03	any further and this is our final simplified answer .
15:09	Now I do want to mention that we always put
15:11	the number before the square root symbol , as you
15:13	can see in all four of our final simplified answers
15:17	. So for example , if we were to put
15:20	the three first , so the square root of three
15:24	Times the square root of 25 and get the square
15:28	root of three times five , you would want to
15:34	rearrange this . So you have the number first and
15:37	put the square root second . So you can see
15:42	that we have the number then the square root .
15:44	So that's common practice and that's how you would want
15:47	to leave your answers . So there you have it
15:50	. There is a general overview of square roots .
15:53	So what are square roots ? Finding the square roots
15:56	of perfect squares and then simplifying square roots . Now
16:00	I would suggest really knowing the 1st 12 perfect squares
16:06	. This is going to help a lot as far
16:08	as square roots go And simplifying them . I have
16:12	a pinned comment below with the 1st 12 . So
16:15	if you need some help with those , check that
16:18	out . I hope that helped . Thanks so much
16:21	for watching until next time . Peace . Mhm mm
16:30	. Yeah .

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