Up next in 10
Operations With Radical Expressions
0 views
Apr 9, 2025
In part 1 of this section, we'll learn how to simplify radical expressions and also learn how to add, subtract, multiply and divide radical expressions.( 6.5) Chapters: 00:00 Introduction 02:21 Using The Product property to simplify radicals 14:59 Using the division property to simplify radicals 17:34 Rationalizing the denominator
View Video Transcript
0:00
you already know your stuff is tomorrow
0:01
i don't need to remind you of that right
0:03
we did we have two days of straight
0:07
uh doing your your preview so we're not
0:10
going to do that today we're going to
0:11
move on and talk about operations with
0:14
radical expressions all right so I gave
0:17
these two as a way to like warm you up
0:19
real quick okay so if I have uh the
0:22
fourth root of 64 x to the 8 y to the 20
0:27
based on what we learned what do you
0:28
think that's going to give you
0:31
x
0:33
uh it's not four this is two here two be
0:38
what 2x 2 x
0:43
2 y y five right 2x 2 x to the 4 on 2x y
0:50
five and then here um square of 25
0:53
x to the 10 so what what I get
0:57
5x to the what to and then what what
1:00
else is that it that's it oh and then um
1:03
because it's a absolute value absolute
1:06
value because we have odd number right
1:10
excuse me the y absolute value the y Oh
1:14
yes that was that was Yeah the y should
1:16
also have absolute value if you recall
1:19
the y should have absolute value right
1:22
so now as you can see for the most part
1:24
with this the first section of things
1:26
that we did every single one of those
1:29
numbers like it was you could actually
1:31
divide like here we did 10 divided by
1:33
two we got five right here cub four root
1:37
of x to the 8 we just 8 / 4 we got two
1:39
so it was always working out for us
1:41
because we always got like uh numbers
1:43
that would divide into the exponent or
1:45
the index but now we have a different uh
1:48
thing here now I have square of 32x
1:51
x to the 8 right and now I know for a
1:55
fact that this is called we're going to
1:57
be talking about using the product
1:59
property to simplify right so now I have
2:02
square root of 32 x to the 8 so what do
2:05
you think is going to happen here 32 is
2:07
not a perfect square so how does that
2:10
change this whole entire operation yet
2:12
do you split up the square we can we're
2:14
going to split them up right we're going
2:16
to split them up or I mean we don't
2:18
necessarily have to split them up but we
2:20
can do that right so if I have 32 I'm
2:23
going to try to break down 32 into a
2:25
number that can give me a perfect square
2:26
right so it's 16 right so I have two and
2:30
16 so I'm going to rewrite this as 16
2:34
right
2:35
* 2 * x to the 8
2:42
right so this is 16 * 2 * x 8 so here
2:46
now I can solve this because now I do
2:50
have a perfect square and I and I do
2:53
have like two I also have a monomial
2:55
that can be broken down yes sir when you
2:58
say 16 * 2 * x 8 would the x go to two
3:04
or would it go it doesn't matter it's a
3:06
multiplication all right so it doesn't
3:07
matter where it goes right cuz I'm
3:09
multiplying multiplication is what
3:11
commutive you can switch the the
3:12
position it doesn't hurt you right so we
3:15
know that 32 is 16 * 2 so now I can find
3:20
what square root of 16 four right so I'm
3:24
start with a form i'll start with one
3:25
that I know I can work on what about x
3:27
to the 8 x to the 4th x to the 4 now
3:31
what do I do with a two that cannot be
3:33
moved right so I have to keep it under
3:37
the radical right so that gives me 4 x
3:40
to the 4
3:41
square root of two so your job is to be
3:44
able to recognize which of those
3:46
monomials can be broken down and if you
3:49
can they cannot be broken broken down
3:50
you have to split them in a way that you
3:53
can get to a result just like this all
3:55
right now it's easier when you just have
3:58
square root now what do we have here now
4:00
we have the fourth root of 16 x to the
4:05
24 and then y the 13 right not only do I
4:08
have numbers I also have like uh
4:11
variables that are broken down and in
4:13
this case here 13 is not a even number
4:18
uh don't don't sit like that anymore
4:19
like just sit regularly because this
4:21
causes a disruption so going forward I
4:25
don't want you to sit like that again
4:26
only on that wall because that's not the
4:27
first time
4:29
[Music]
4:30
right so here we're going to break down
4:33
the 16 right i know that what's What's
4:36
uh the fourth root of 16
4:39
two right two cuz two * 2 * 2 actually
4:43
this one was wrong here this should have
4:45
been four i just caught that i said four
4:48
i did say four too none of us like right
4:54
uh but you can't break up to 13 right i
4:57
can't break the 13 but I could do
4:58
something here right so now here right
5:01
guess what happens we have four of 16 we
5:05
know that this is going to be actually
5:07
I'm going to rewrite this right i'm
5:09
going to rewrite it first before I even
5:10
solve it so let's rewrite this to to
5:13
kind of like make it easy for oursel we
5:15
break down 16 if I break down 16 is four
5:18
and four right and then four is two and
5:21
two two and two so I can write 16 as two
5:25
to the power four i won't forget to put
5:27
my four in my index right and now I know
5:31
x is 24 i know that four goes into 24 so
5:34
I don't need to do anything to it okay
5:36
i'm not going to do anything to it so
5:38
I'm going to leave it alone as 24 now 13
5:42
I'm going to break down right cuz I know
5:44
13 can be broken down as what y 12 * y
5:48
because I know that 4 goes into 12 so
5:50
I'm going to break it down in such a way
5:52
that it'll be easy to simplify so now
5:55
this is going to be y 12 * y right why
5:59
because it's y one okay so now we have
6:02
an expression that can be
6:06
simplified is are you following we good
6:09
but it's 16 is a perfect square so I
6:12
would just I'm Yeah I'm going to get to
6:14
that i'm getting to that i'm just going
6:17
I'm just going baby steps i don't want
6:20
to rush it it's because there's a number
6:21
four on the side so you're not finding
6:23
the square you have to find find the
6:24
four roots right but I'm going step by
6:26
step again I broke down this because I
6:28
want to match up the exponent the index
6:30
of the exponent so I can simplify it
6:32
okay does that make sense not really so
6:36
where do I Did I lose you why do we have
6:39
to add the four
6:42
what do we have to four here i just
6:45
broke down 16 to 2 to the^ 4 right so
6:48
that way I know that the four root of 16
6:50
is two but to make it understandable I
6:52
want to break it down to show you why
6:54
this is actually two right because we do
6:58
what four divided by what four which is
7:01
one which is two right so that's how we
7:03
got what we got but I want to show you
7:05
the simple steps so you can see it right
7:08
so now I'm going to bring it down here
7:10
so now I have the 4 of 234 * x 24 * y 12
7:17
and then y so I'm going to start taking
7:19
out things that are simplifi can be
7:21
simplified right so what's the fourth
7:23
root of two to the four two right that's
7:26
going to be two what's the fourth root
7:28
of x to the 24 x to the 6 x to the 6 and
7:32
what's the fourth root of Y to the 12 y
7:35
y cub and I'm going to have the absolute
7:38
value right and then the only one that's
7:39
left inside is what y four of Y right so
7:44
that will still be left inside of it
7:47
right that cannot be because y cannot be
7:50
simplified so I have to leave it under
7:52
the
7:53
radical did we get it or do I need to go
7:56
over this a second can you go excuse me
7:59
can you do another problem yeah yeah I
8:01
have another problem here right so we
8:04
have now we have let's say we have this
8:06
one we have 12 right square root of 12 x
8:10
to the
8:11
6 and then y cubed so what am I going to
8:15
do here this is a square root so this is
8:17
a lot easier to work with right which
8:19
number do I need to break down which one
8:20
do I need to leave alone you need to
8:23
break down 12 and y to the cub and you
8:25
leave alone 8 to the 6 that's right
8:27
because this is even so I don't need to
8:28
work on it i need to leave it alone
8:30
right but I'll break down 12 12 will be
8:33
4 * 3 because I know now I have a
8:35
perfect square right so I'm going to
8:37
rewrite this
8:39
as 4 right * 3 * x 6 and I'm going to
8:45
break down y cub into y^2
8:48
* y you see why we're doing
8:51
this so we want to find the square root
8:54
right so square root means I need to
8:57
find perfect squares you know what a
8:59
perfect square is right right because 12
9:01
without a perfect square so I can't just
9:04
find the square root of 12 i need to
9:06
find a perfect square because I'm trying
9:07
to simplify the expression right so I
9:10
know I can break down 12 and I can
9:12
rewrite this as 4 * 3 because now I have
9:15
a perfect square in it okay now x 6 has
9:19
to be left alone because I know I can
9:21
find the square root of x to the 6
9:24
because this exponent is
9:26
even right and then y to the3 power that
9:31
also is not is not an even number so I
9:34
need to break it down into y^2 * y okay
9:38
now once you break it down now you can
9:39
start taking each one of them
9:41
individually and find the square root
9:44
right now what's square root of four two
9:46
that is out right and now square root of
9:49
x to the 6 x the 3r x the third we're
9:52
just going to have the absolute value
9:54
right and then square of y square y
9:57
that's also going to be an absolute
9:58
value so I can I can actually put these
10:00
two in the same absolute value right and
10:03
then the last one what is left inside
10:06
the only thing left inside is square
10:07
root of 3 y okay so all you're doing is
10:11
just finding
10:13
uh those numbers and exponent that can
10:18
be simplified and everything else that
10:20
cannot be simplifi simplified stays
10:23
under the radicals okay so I'm going to
10:26
continue to work on these i'm going to
10:27
sit on that for a little bit because I
10:29
do believe this is a little bit more
10:32
complicated than the last thing that we
10:33
did so can I erase this and give you a
10:35
couple more problems you have to work on
10:37
okay
10:39
can I have a eraser
10:41
yeah one second can I have one
10:45
can you get erasing marker uh yeah got
10:48
that a marker
10:53
i don't have any more markers put one
10:55
more on the board right there oh oh yeah
10:57
yeah there's one here
10:59
all right so let's assume that we have
11:04
Yeah give me one second
11:06
all right so say I have uh
11:13
144 right x is 7
11:17
Y to the 9 right and then I have
11:25
um Oh huge numbers here you know one
11:32
yeah I'm just No I'm looking at like
11:34
some similar problem
11:37
so all right we got that and then I have
11:41
cub
11:52
all right
11:54
so I have these two here right i have
11:57
100 square of 144 x to the 7 and
12:02
then y to the 9 how do you think we're
12:04
going to work on this so which are the
12:07
perfect squares and if it if there's no
12:10
perfect um there's no in this case even
12:15
x and what I do have 12 right 124 12 so
12:20
we can take that out right but we can go
12:22
step by step we need to still break down
12:23
x to the 7 how are we going to break
12:25
that down you know x^ 6 * x x 6 * x
12:29
that's excellent and then what else y 8
12:33
* All right that's also good y * y and
12:36
now what do we have
12:39
if I was to find the square root of 12
12:42
12 x to the
12:46
x absolute value and then y y to the
12:51
fourth okay no absolute value here
12:54
needed you have x and y under that's it
12:58
x and Y under right so all you're doing
13:00
is just really just simplifying that
13:02
good now what about that here so what do
13:05
we do here
13:07
so now I have 36
13:14
square root of x is six remember when we
13:17
say square root we say that there's an
13:18
imaginary what here two right so this is
13:22
by powers of two so the index is two so
13:25
this is why we got x to the 3 right okay
13:28
now we have cubic root of 36 x to the 5
13:32
y to the 4 can I do anything to the 36
13:37
well if you still 12 and three 12 and
13:40
three it still won't help right no nine
13:43
and four yeah nine and four that still
13:45
won't do anything right three and three
13:47
three and three that'll be three * 3 so
13:49
I I have to leave the 36 alone because
13:51
there's no number here that's going to
13:52
give you right no cubes for it no cube
13:55
root for it so I leave that alone so she
13:58
could be misleading sometimes so 36 but
14:01
then I could work on x to the 5 break it
14:03
down into what three and two two right
14:06
because I'm I'm looking at a cubic root
14:07
here and then x y 4 be y3 and y right so
14:13
now I can take the cubic root of x the 3
14:16
that gives me x no need for an absolute
14:20
value because this is a this is a odd
14:22
number so I do not use the absolute
14:25
value here right i don't use it so you
14:27
guys got to remember that and then here
14:29
we got y and then left hand side we have
14:32
36
14:34
uh x² and y and this is it so this is
14:37
why you really have to watch the index
14:39
which is this number here because if you
14:42
do not watch the index it may screw your
14:46
operational all right
14:52
all right so now we can move on to the
14:54
quotient property quotient property
15:04
you good all right are you sure
15:09
so now we're going to talk about a
15:10
quotient property all right quotient
15:12
property so what is it so it basically
15:15
tells us that if I have a big radical
15:19
covering two monomials that are being
15:21
divided then I can split them into two
15:24
right so basically the n root of x y is
15:29
the same as me saying the n root of x
15:32
over the n root of y example if I have
15:36
cube root of 27 x cub 8 that means this
15:41
is the cube root of 27 x
15:44
cub over the cube root of 8 by itself
15:47
you can split them into two right you
15:50
can do that and then simplify your um
15:54
expression so the quotient property is
15:57
basically telling you if you have a big
15:59
radical covering two
16:03
uh monomials you can split them into two
16:05
separate ones okay so that's all it is
16:08
so let's say I wrote like square root of
16:11
16 over 5 you can say oh this is square
16:14
root of 16 over five that's what this
16:17
rule means okay you can split that into
16:20
two that that's all it is that's a
16:22
quotient property now it comes in handy
16:24
when we come when it comes on to like
16:26
rationalizing the denominator all right
16:29
and so we're going to learn how to do
16:30
that
16:33
here all
16:36
right now this is where I'm going to
16:38
need you to really
16:40
like I mean it might be hard for some it
16:42
might be easy for some so I don't know
16:44
we'll see so here I want to rationalize
16:48
the denominator what does that mean i
16:50
want to rationalize the denominator
16:52
what's the expression rationalizing the
16:54
denominator mean
16:57
we did that in your we did that like
16:59
right here means making the denominator
17:02
not have
17:03
a square that's right right that means
17:07
we want to get rid of the square root uh
17:10
symbol in the denominator when you hear
17:12
the word rationalize something that
17:14
means you want to get rid of the square
17:16
roots right but before I do that I need
17:19
to apply the quotient property right so
17:22
this is a square root here is covering
17:24
both the x to the 6 and the y to the 7
17:27
so now I'm going to split them into x to
17:30
the 6 like this and then y to the 7 and
17:35
then I'm going to try to rationalize
17:36
this right so first before I even
17:39
rationalize it can I simplify the top
17:41
so what is that going to give me
17:45
absolute absolute value right because
17:48
this square root this and then you have
17:51
an odd number here and then now we have
17:53
to work on y 7 now watch this here this
17:56
is y to the 7 right so what do I need to
18:00
do to rationalize this i want to get rid
18:03
of the radical so what can I do here
18:04
yeah multiply by what is your uh square
18:08
root of y square root of y why is this
18:12
square root of y and nothing
18:14
else because this is y 7 if this turns
18:18
into a 8 can I simplify right so that
18:21
means I can do that so I'm going to
18:23
multiply this by what square root of y
18:27
and then square root of y because when I
18:29
do something in the denominator I have
18:31
to repeat it in the numerator you can't
18:33
do for one and and then and then ignore
18:36
the other it has to be done because we
18:38
are trying to rationalize this right
18:40
they're trying to rationalize mean we
18:42
are trying to get rid of the radical but
18:43
to get rid of this radical all I have to
18:45
do is add one more y to it cuz if I have
18:48
y to the 8 now it's a even number i can
18:51
take the square root of that even number
18:53
that's going to give me y to the 4 right
18:56
that's the whole goal is to rationalize
18:58
the denominator we are not rationalizing
19:00
the numerator but the denominator right
19:03
so that that means now we're going to
19:05
have absolute value of x cub y over y^ 8
19:12
and now we can simplify
19:14
this because if you simplify this that
19:17
gives you y 4 so now you rationalize it
19:20
so you get x
19:21
cub y over y 4 right so this is how
19:27
you rationalize the denominator when
19:29
it's just a simple square root
19:31
it's going to get complicated when we do
19:33
the same thing for something like this
19:36
which has like a foroot so let's let's
19:38
work on that okay we're going to do a
19:40
couple of those the principle is the
19:42
same it's not hard it's the same
19:44
principle i'm going to give you some
19:46
little ones to start with and then we're
19:48
going to work on the bigger ones so
19:50
let's say I have um square root of
19:53
five over is that um let's
19:58
say uh x
20:02
uh x cub and I want you to rationalize
20:06
the denominator how you going to do that
20:09
first you split them up okay
20:12
[Music]
20:15
okay and then what and then
20:19
you square x both nominator okay why is
20:23
this not square why why is it square of
20:25
x and and not square of x² because um it
20:30
to two to two right we want to get this
20:33
three and one is four all right so now
20:36
we have square root of 5x here right and
20:39
then under here what we going to get
20:42
x to the two so which is well four is
20:46
still is still under the radical and
20:48
then when you simplify it it gives you
20:49
what
20:51
x right oh this is 12 this is three okay
20:55
yeah this is three right now let's let's
20:58
let's let's stretch it a little bit now
21:00
what if I had Wait isn't x Oh wait no
21:04
it's never mind i figured it out so what
21:06
if I had uh like
21:11
this i have y
21:15
uh y 6
21:18
over x to the
21:22
x to the two right and I want to
21:24
rationalize the denominator what do I do
21:26
here
21:29
what do I do actually I make it a little
21:32
harder than that that's too easy
21:34
so let's say we have x uh just x yeah x
21:40
to the 7 right x to the 7 and I want to
21:43
rationalize the denominator so what am I
21:46
going to do here first split them split
21:48
them all right what do we get
21:51
we get
21:54
y^ 6 and then
21:56
cubet x now how do I rationalize this
21:59
here would
22:02
you yeah x2 why x2 because you're trying
22:06
to get it to 9 to 9 so we can match it
22:08
up right so it's going to be x squ on
22:11
both sides right
22:13
x2 and then x2 here right and now we
22:17
have this we we know we can simplify
22:20
this can't we
22:22
cube root of y 6 is what y the second y
22:25
the 2 right and then it's going to be
22:27
cube root of x² and then under what are
22:31
we going to get
22:33
x the 3r x the third absolute value x 3r
22:37
absolute value oh no because it's a cub
22:39
so it's just x the 3r right so that'll
22:41
be x the third here because this will be
22:43
cubet of x 9 and then 9 / 3 is 3 so x
22:48
the third right okay now we have that
22:54
yeah question
22:59
well the thing here right
23:02
the bottom right
23:04
so this time this gives you what cube
23:06
root of x to the what
23:10
7 + 2 9 nine right and now we can
23:14
simplify the bottom what's cube root of
23:16
x of 9
23:18
x the third right and then we have
23:20
whatever we have here oh okay occupy
23:23
this space okay now I'm going to try and
23:25
work on this one i have four root of
23:30
6 over
23:33
5x i have a four root of 6 over
23:37
5x so what do I do here
23:42
them all right let's
23:46
And then
23:49
you times it by x to the 3 or x 4 all
23:53
right I also have to worry about what
23:55
the number right so think we have two
23:57
things here we don't we don't only have
23:59
the the the
24:01
we have also the five right so I'm going
24:03
to multiply by what the bottom one
24:08
say it again 3x this is five so how many
24:12
how many fives did I need to add to this
24:13
five three three fives will be five to
24:16
what third right and then x to the
24:22
third why is it 5 to the third and x 3r
24:25
5 to the 4th 5 to the 4th right cuz this
24:28
is 5x five is like 5 to the one this is
24:31
x to the one so to rationalize it I need
24:34
to add three more fives to the five and
24:38
I need to add three more x's to the x in
24:41
terms of
24:42
multiplication right are we good on that
24:46
all right so we are trying to
24:48
rationalize the denominator right
24:50
meaning I am trying to get rid of
24:54
the I'm trying to get rid of the radical
24:57
under here
24:59
right i'm trying to get rid of this
25:02
radical so this is five and this is x
25:07
okay now what if hypothetically speaking
25:10
right this was 5 to the 4 and x to the 4
25:13
so what would you get for the bottom
25:16
just 5x 5x right but now that's not what
25:20
you have now you have this
25:23
5x so but I want you to get to that 5x
25:26
without the right so how much more do
25:28
you need to add to this and this to get
25:29
to that three three right so this is why
25:32
this comes in handy now we have we have
25:35
to add 5 to the 3 and then x the 3r and
25:38
then we have to do the same thing up top
25:41
right
25:43
we have to do the same thing up top
25:44
because anytime you do something in the
25:46
denominator you have to do the same the
25:49
six alone you just leave the six alone
25:50
because it's irrelevant we just 750 does
25:54
not have a cube root right so now if I
25:57
do this now I get 4 of 5 to the 4th x
26:03
4th And then over whatever this is here
26:05
right someone trying not to put this in
26:07
the calculator what's 6 * what's 5 to
26:10
the 3r 5 125 right yes yeah 125 * 6 is
26:15
750 okay 750 50 50 so 750 x cub and then
26:20
now you can rationalize the denominator
26:23
right so you're left with
26:27
750 x cub over 5x and this
26:31
is where he's trying to go with it this
26:34
is your your um final result for the sec
26:38
for the rationalizing is that it no
26:41
there's more stuff here but I'm thinking
26:44
about pausing yeah can we can we just
26:47
focus on this all right
26:51
i already gave you two days for the quiz
26:52
to review so I give too much let me see
26:58
oreo m can you just at least give us
27:01
more practice of this yeah yeah let's do
27:03
let me do that let me pause you know