Math Antic - Simplifying Square Roots - Free Educational videos for Students in K-12

Math Antic - Simplifying Square Roots - By mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math Antics in this lesson , We're going to learn
00:09	a bit more about roots and how you can simplify
00:12	them . Have you ever noticed that teachers can be
00:14	kind of picky sometimes ? Like if you give them
00:17	an answer that's not in the form they want ,
00:19	you might lose a point even if the answer is
00:22	technically correct . That's partly because in math there are
00:25	many different ways to write the exact same number or
00:28	expression , but some are much more clear and helpful
00:31	than others . Take the number one . For example
00:33	, you can write it as the fraction 2/2 or
00:36	you could write it as the expression one plus zero
00:39	. You could even write it as the square root
00:41	of one if you wanted to . The possibilities are
00:43	endless . But if it's the answer to your problem
00:46	, why would you use any of these more complicated
00:48	forms when you could just write one ? Isn't that
00:51	a lot simpler ? It sure is . And math
00:53	teachers always like it when answers are written in the
00:56	simplest form possible . So let's suppose you do a
00:59	math problem and get the answer the square root of
01:02	16 , You can just leave it like that and
01:04	it wouldn't be wrong . But can you think of
01:06	a simpler way to write it ? Of course 16
01:09	is a perfect square , it equals four times four
01:12	. So the square root of 16 is just four
01:15	. That's definitely a simpler answer . But what if
01:18	you do a different problem and end up with the
01:20	answer ? The square root of 32 ? Well 32
01:24	isn't a perfect square . So we can't simplify it
01:27	to a nice hole number . Like we could with
01:28	the square root of 16 , you might consider using
01:32	a calculator to convert it to a decimal value .
01:34	But in this case we would end up with an
01:36	irrational number , which is a never ending , never
01:39	repeating decimal and that's definitely not a simpler way to
01:42	write it . So what other options do we have
01:45	? Well , some of you may notice that 32
01:48	could be factored into 16 times too . Right .
01:51	And we already know that 16 is a perfect square
01:55	. But does that help us out ? Actually ,
01:57	it does because of a particular rule about square roots
02:00	. That rule says if you have to square roots
02:03	that are being multiplied together , like the square root
02:05	of two times the square root of three , you
02:08	can combine them like this , the square root of
02:10	two times three and you can go the other way
02:13	too and un combine them . So if you start
02:15	with the square root of two times three you could
02:18	change it to the square root of two times the
02:20	square root of three . We can use that rule
02:23	to our advantage in our current problem because we just
02:26	figured out that the square root of 32 is the
02:28	same as the square root of 16 times two ,
02:31	which means we could rewrite it as the square root
02:34	of 16 times the square root of two . And
02:37	as we already know , the square root of 16
02:39	can be simplified to just four . So that gives
02:42	us four times the square root of two or just
02:44	four route to as a simplified version . Pretty cool
02:47	. Huh ? Hold on a second . How is
02:50	that answer ? Simpler than what you had before Before
02:53	you just had the square root of 32 and now
02:56	you have a whole number times a route that seems
02:58	even more complicated to me . Yeah , I know
03:01	what you mean . Sometimes it's hard for people to
03:04	decide what the simplest or most helpful form is .
03:08	So mathematicians rely on what are called conventions where they
03:12	all agree on a preferred way to express things .
03:14	Ooh that sounds fun . A convention for math .
03:18	Like math con Hey hey hey , would you like
03:24	a selfie with the math antics guy ? Oh that
03:26	that's okay . Thanks . Anyway , I'm actually looking
03:29	for the khan academy booth . Oh there it is
03:33	. Mhm . Gone . Gone . Well it's true
03:40	that the word convention often refers to a group of
03:43	people meeting together in a big building . But when
03:46	I say convention , I mean it's a conventional or
03:49	standard way of doing something . Hey , want to
03:52	see a picture of me and sal khan . Anyway
03:55	in math it's conventional to simplify a route . If
03:58	you can simplifying a route means identifying any factors under
04:03	the radical sign that if you took the route of
04:05	them would simplify to a whole number , which you
04:08	could then bring out in front of the radical sign
04:11	even though the result might seem less simple than before
04:14	. If the part under the radical sign has been
04:16	made as simple or as small as possible , then
04:19	you've simplified the route . This definition of simplifying would
04:23	apply to any type of root , square roots ,
04:25	cube roots , fourth roots and so on before the
04:28	next couple examples will just focus on square roots since
04:32	they're the kind you'll encounter most often in the case
04:35	of square roots . To simplify , you'll need to
04:37	find out if there's any perfect squares hiding in the
04:39	number under the radical science . You remember what a
04:42	perfect square is , right ? It's just what you
04:44	get when you multiply a whole number by itself ,
04:47	like two times two or five times five or 30
04:50	times 30 . And how would you find out if
04:53	there are any perfect squares hidden in that number ?
04:55	The key is to factor it . If you've forgotten
04:57	how factoring works , you can watch our previous videos
05:00	about it for help factoring will reveal if there's any
05:03	pairs of the same factor being multiplied together Like two
05:06	times 2 or three times 3 If there are you
05:09	found a perfect square hiding in that number and you
05:12	can simplify it . Like in this example the prime
05:15	factor ization of 180 is two times two times three
05:19	times three times five . So you can rewrite it
05:22	like this and look for perfect squares two times 2
05:26	is a perfect square since its value is four and
05:29	three times three is a perfect square since its value
05:31	is nine . Now remember the rule I mentioned earlier
05:34	that says you can rewrite a square root as a
05:37	product of the square root of its factors . That
05:40	means we can rewrite the problem like this . Do
05:43	you see the advantage of doing that now we can
05:45	simplify the square roots of the perfect squares that we
05:48	found so that they just become whole numbers . The
05:52	square root of two times two simplifies to two and
05:55	the square root of three times three simplifies to three
05:58	. That gives us two times three times the square
06:00	root of five . The square root of five can't
06:03	be simplified any further . five is a prime number
06:06	. So it's only factors are one and itself great
06:09	. Now all we have to do is recombine the
06:11	factors , we simplify it two times three equals six
06:14	. So the simplified version of the square root of
06:16	180 is six times the square root of five or
06:20	six . Route five . Let's try another example to
06:23	make sure you understand what's happening , Let's simplify the
06:26	square root of 72 . If we factor 72 all
06:29	the way down to its prime factors , we get
06:31	two times two times two times three times three .
06:34	As you can see , there are two pairs of
06:37	factors that form perfect squares two times two and three
06:40	times three . And there's a two leftover that doesn't
06:43	form a pair . Like before we could rewrite this
06:46	using our rule about multiplying roots . But now that
06:49	you know how that rule works , you can eliminate
06:51	some of the in between steps . If you realize
06:54	that any pair of identical factors that are under the
06:57	square root sign will simplify to become a single factor
07:00	out in front of the root sign . That means
07:03	you can just change the two times two under the
07:06	root sign into a two out in front of it
07:09	and the three times three to a three out in
07:11	front . That gives us two times three times the
07:14	square root of two or six route to . As
07:16	a simplified answer . So that's basically all there is
07:20	to simplifying square roots . And even though square roots
07:23	are the most common , sometimes you may need to
07:25	simplify other routes to , For example , what if
07:28	you need to simplify the cube root of 72 instead
07:31	? Well you would start the process the same way
07:34	you factor the number under the root . Sign down
07:37	to its prime factors to see if any parts of
07:39	it can be simplified . But since we're working with
07:42	a cube root this time , that means we aren't
07:44	looking for pairs or perfect squares anymore . Instead we
07:48	need to find factors that are in groups of three
07:50	. In other words , we're looking for perfect cubes
07:53	just like before 72 factors down to two times two
07:57	times two times three times three . But since we're
08:00	dealing with a cube root this time , the two
08:02	times two times two can be simplified because it's a
08:05	perfect cube , but the three times three can't because
08:08	it's a perfect square . Also , like before we
08:11	can use our rule to break this problem up into
08:14	the cube root of two times two times two times
08:17	the cube root of three times three . This first
08:20	part simplifies to two . Well , the second part
08:23	can't be simplified any further , so we just recombine
08:26	it to the cube root of nine there . The
08:28	simplified version of the cube root of 72 is two
08:32	times the cube root of nine . So when simplifying
08:35	roots in general , it's important to pay attention to
08:38	which kind of route you're dealing with . So you
08:40	know , which factors can be simplified and which can't
08:43	basically , you just look at the index number and
08:46	then try to find groups of identical factors of that
08:48	size for square roots . It's groups of two for
08:52	cube roots , groups of three for fourth roots ,
08:54	it's groups of four and so on . Hopefully that
08:57	all makes sense . But if you're still having trouble
09:00	getting it , I'd highly recommend re watching our videos
09:03	about exponents and routes because it's super important to fully
09:07	understand how they work before you can understand how to
09:10	simplify them . Oh , and there's one more quick
09:13	thing that I want to mention in this video that
09:15	has to do with conventions about roots . Sometimes an
09:18	answer to a math problem might be a fraction with
09:21	a root in the denominator like this . one ,
09:23	three over the square root of two . The square
09:26	root of two can't be simplified . But many mathematicians
09:30	like to avoid having roots in the denominator , especially
09:33	when those roots are irrational numbers . So if your
09:36	teacher asked you to rewrite this fraction without a root
09:39	in the denominator , how would you do it ?
09:41	Well , think back to when you learn how to
09:44	add or subtract . Unlike fractions , which means fractions
09:47	that don't have the same denominator . In those cases
09:51	you needed to change the fractions to like fractions by
09:54	multiplying one or both of them by a special hole
09:57	fraction . We can apply that same idea to change
10:00	the denominator of this fraction into a regular whole number
10:03	instead of a route . All we have to do
10:05	is make a whole fraction out of that radical denominator
10:09	. In this case that would be square root of
10:11	two over square root of two . Then we multiply
10:14	our original fraction by that new whole fraction whose value
10:17	is just one on the top . We would get
10:20	three times the square root of two on the bottom
10:22	. We would get squared of two times square root
10:24	of two , which by definition will just equal to
10:28	. So these two fractions are equivalent . They represent
10:31	the exact same value , but one has a radical
10:34	in the denominator while the other has a radical in
10:37	the numerator , neither answer would be wrong . But
10:40	if there has to be a radical or a root
10:42	and a fraction , then mathematicians prefer to have it
10:45	in the numerator as a convention If you'd like to
10:48	research this idea further , it's often called rationalizing the
10:52	denominator because it's a way to change a fraction so
10:55	that it has an irrational numerator instead of an irrational
10:58	denominator , which makes some mathematicians feel much better .
11:02	All right , that's all for this video . Hopefully
11:06	it's given you a better understanding of some of the
11:08	conventions in math concerning roots . Knowing about them can
11:11	come in handy on tests in particular . For example
11:14	, if you solve a problem and get the answer
11:17	square root of 72 , but the multiple choice answers
11:20	available to you are all in simplified form , like
11:23	six route to you'll realize that you just have to
11:25	simplify your answer so you know which one to pick
11:29	and remember Math is a subject that you can't truly
11:32	learn just by watching . You actually have to apply
11:34	it . So be sure to practice simplifying routes on
11:37	your own . As always . Thanks for watching Math
11:39	antics and I'll see you next time . Hey ,
11:43	you wanna selfie with the Math Antics guy ? Learn
11:56	more at Math antics dot com .

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