in this video we're going to focus on
how to simplify radicals with variables
and exponents so let's say if you want
to simplify the square root of x to the
fifth the index number is a two now one
way you can do this is you can write x
five x and because there's a two you
need to take out two at a time so this
will come out as one X and this will
come out as another X and you're going
to get x times X square root of one at
just X by itself so this is equal to x
squared root X now another way you can
simplify this or get to the same answer
is if you do it this way how many times
is to go into five two goes into five
two times because two times two is four
two times to be at six that's too much
and what's remaining 5 minus 4 is 1 so
you get one remaining that's another way
you can simplify it so let's say for
example if you want to simplify the
square root of x to the 7 how many times
does 2 go into 7 2 goes into 7 three
times with one remaining now let's try
this one
how many times does 2 go into 8 2 goes
into 8 or 8 divided by 2 is 4 2 goes
into 8 4 times with no remainder 2 goes
into 9 4 times and 2 goes into 12 6
times or no remainder for the 9 there's
a remainder of 1 so the Y is still on
the inside that's a quicker way that you
can use to simplify our radicals now
let's say if you have a number let's say
if you want to simplify the square root
of let's say 32 what you want to do is
break this down into two numbers one of
which is the perfect square so 32 can be
broken down into 16 and 2 now the reason
why I chose 16 is n 2 is because we know
what the square root of 16 is and that's
4 and so this is just 4 root 2 now let's
say if we have a problem that looks like
this let's say if we want to simplify
the square root of 50 X cubed Y to the
18th
so how many times does 2 go into 3 2
goes into 3 one time with one remaining
and 2 goes into 18 nine times now
usually when you have an even index and
an odd exponent you got to put it in
absolute value now your teacher may not
go over this but some teachers do but
just in case if you have one of those
teachers who wants you to use an
absolute value you only need it if you
have an even index and if you get an odd
exponent after it comes out of the
radical now the only thing you have to
simplify is root 50 square root 50 we
can break it down into the square root
25 and 2 because 25 times 2 is 50 and
the square root of 25 is 5 but the 2
stays inside the radical so we can put a
5 on the outside and let's put the 2
inside so this is the final answer
that's how you can simplify that
expression let's try some other problems
so let's say if we have the cube root of
x to the fifth Y to the knife and Z to
the 14th so how many times does 3 go
into 5 3 goes into 5 one time with two
remaining so we're going to put x
squared inside and the index number wood
is going to stay 3 now how many times
does 3 go into 9 9 divided by 3 is 3
with no remainder and how many times
does 3 go into 14 3 goes into 14 4 times
and 3 times 4 is 12 so 14 minus 12 is 2
so we have 2 remaining so that's how you
can simplify radicals let's try one
final problem feel free to pause the
video and see if you see if you can get
the answer for this one so the cube root
of 16 X to the 14th Y to the 15th Z to
the 20th
so how many times does 3 go into 14 3
goes into 14 4 times 3 times 4 is 12 and
14 minus 12 is 2 so we're going to get x
squared on the inside 3 goes into 15 5
times with no remainder because 15
divided by 3 is 5 3 goes into 20 6 times
3 times 6 is 18 3 doesn't go into 20
evenly and 20 minus 18 is 2 so there's 2
remaining now let's simplify the cube
root of 16 perfect cubes are 1 1 cube is
1 2 to the third power is 8 3 to the
third power is 27 so a perfect cube that
goes into 16 is 8 so 16 divided by 8 is
2 so you want to write cube root of 16
as the cube root of 8 times cube root or
two because the cube root of 8
simplifies to 2 so this 2 is going to go
on the outside which we're going to put
it here and this 2 remains on the inside
which I'm going to put in there so this
is our final answer for that problem so
that's how you can simplify radicals
with variables and exponents but
actually let's try one more let's say if
you have a question it looks like this
let's say the square root of 75 X to the
7th Y to the 3rd Z to the 10th over 8
let's say X to the third Y to the knife
Z to the fourth so the first thing we
can do is um let's simplify everything
let's rewrite it so seventy-five is 25
times three we can square root 25 that's
five but we'll do that later and eight
is four times two because we can take
the square root of four now when you
divide exponents I mean when you divide
variables you got to subtract the
exponents 7 minus 3 is 4 and that goes
on top because there's more x-values on
top now for this one you can do 3 minus
9 but I think it's easy if you
subtracted backwards 9 minus 3 which is
6 and because we subtracted backwards
the 6 goes on the bottom and then 10
minus 4 so there's more Z's on top down
the bottom so we're going to put it on
top so Z to the 6 and now let's simplify
the square root of 25 is 5 and a square
root of 4 is 2 now 2 goes into 4 2 times
4 divided by 2 is 2 so we get x squared
2 goes into 6 3 times so we get Z to the
third and 6 divided by 2 is 3 so we get
Y to the third and inside the radical we
still have a radical 3 and the square
root 2 leftover so now we also need to
add some absolute values because we have
an e to even index and we have a few odd
exponents we need to put Z and absolute
value and a y so our last step is to
multiply top and bottom by square root
of 2 we need to rationalize the
denominator we need to get rid of that
radical so our answer in our final
answer is 5 x squared
absolute value of Z to the third square
root 6 over
square root two times square root two is
square root of four which simplifies to
two and two times two gives us four so
we get for absolute value y cube this is
our final answer for that particular
problem
okay let's try just one more problem so
let's say if we have the cube root of
sixteen X to the 7 y to the fourth and Z
to the Knife divided by fifty four x
squared Y to the knife is Z to the
fifteenth so feel free to pause the
video and try this example yourself so
the first thing I will do is within a
radical I would divide both numbers by
two referring to the 16 and the 54 so
right now what I have is the cube root
of eight which is a perfect cube over 27
sixteen divided by two is eight half of
54 is 27 so now what I'm going to do is
subtract the exponents 7 minus 2 is 5
and for the Y's I'm going to subtract it
backwards 9 minus 4 is 5 so Y to the 5th
+ 4 z I'm going to subtract it backwards
15 minus 9 is 6 but that's going to go
on the bottom and so now we could
simplify it so the cube root of 8 is 2
and 3 goes into 5 only one time with 2
remaining and the cube root of 27 is 3
and 3 goes into 5 one time just like X
with 2 remaining 3 goes into 6 two times
so that becomes Z squared
now we don't need any absolute values
because this is an odd index we only
needed four even index numbers that
produce an odd exponent so now let's
simplify what we have so we need to get
rid of the radical on the bottom so
we're going to multiply top and bottom
by the cube root of Y to the first power
so what we now have is two X cube root x
squared times y divided by three y times
Z squared times the cube root of Y to
the third the cube root of Y to the
third cancels and so that becomes Y to
the first and Y to the first times y to
the first is y squared so our final
answer is 2x cube root x squared Y over
three Y squared Z squared and that's it
so now you know how to simplify radicals
with variables and exponents so that's
it for this video thanks for watching
and have a have a wonderful day