How to factor out a square root
How To Simplify Square Roots
Timecodes:
in this video we're gonna focus on
simplifying square roots we're gonna
start with some basic examples and then
gradually they're going to get harder so
consider these four examples how would
you simplify it let's start with the
first one what is the square root of 49
what two identical numbers when
multiplied will give you 49 49 is 7
times 7 so it turns out that the square
root of 49 is 7 what about the square
root of negative 25 well this won't give
you a real number but this will give you
an imaginary number what you can do is
break it up into 25 times negative 1 and
it's important to understand that the
square root of negative 1 is the
imaginary number I now what is the
square root of 25 what two numbers
multiplied what 2 identical numbers when
you multiply them will give you 25 we
know that 5 times 5 is 25 so the square
root of 25 is 5 and the square root of
negative 1 is I so this gives us the
imaginary number 5i now what about
negative square root 81 this time it's a
little different than a previous example
the negative is on the outside so we're
not going to get an imaginary number but
we're going to get a real number and
that negative sign will remain on the
outside so what is this square root of
89 I mean excuse me what is the square
root of 81 what number times itself is
equal to 81 now we know that 9 times 9
is 81 so the square root of 81 is going
to be 9 so the answer that we get in
this case is negative 9 now what about
negative
square root of negative 64 what's the
answer there if you have a negative sign
inside a square root it's best to remove
it by writing the square root of
negative 1 next to it so we can replace
this with I now the square root of 64 is
8 because 8 times 8 is 64 and so the
final answer is going to be negative 8i
now sometimes you may have to simplify
square roots that don't contain perfect
squares
for example how can we simplify the
square root of 18 and a square root of
75 now you need to understand what are
perfect squares 1 is a perfect square
because 1 times 1 is 1 4 is a perfect
square
2 times 2 is 4 9 is a perfect square
because 3 times 3 is 9 4 times 4 is 16 5
times 5 is 25 6 times 6 is 36 7 squared
is 49 8 squared is 64 9 squared is 81 10
squared is 100 so these are known as
perfect squares because if you have the
square root if any of these numbers like
let's say the square root of 36 it's
gonna simplify to an integer but now 18
and 75 are not included in its list so
how do we simplify the square root of 18
and the square root of 75 what would you
do here one thing that you could do is
you can break up the number 18 into two
smaller numbers one of which contains a
perfect square so what perfect square
goes into 18 18 is divisible by 9 so
what I would do is write the square root
of 18 as being the square root of 9
times the square root of 2 because 9
times 2 is 18 and now at this point we
know what the square root of
- the square root of nine is three and
so the final answer is 3 square root 2
and so that's a simple way in which you
could simplify square roots let's do the
same for the square root of 75 so what
is the highest perfect square that goes
into 75 25 goes into 75 75 divided by 25
is 3 so we can write 75 as being 25
times 3 and the square root of 25 is 5
so the square root of 75 simplifies to 5
square root 3 now let's work on some
more similar problems for the sake of
practice try these two the square root
of 12 and a square root of 48 feel free
to pause the video as you work on those
two examples so the highest perfect
square that goes into 12 is 4 so we can
write 12 as 4 times 3 and the square
root of 4 is 2 so this is going to give
us 2 square root 3
now what perfect squares can go into 48
48 is divisible by 4 and it's also
divisible by 16 so what are you doing
this scenario if you have multiple
perfect squares that can go into a
number pick the highest one in this case
16 48 divided by 16 is 3 so we can write
48 as being 16 times 3 and the square
root of 16 is 4 so the answer is going
to be 4 square root 3
try these two problems for square root
ninety-eight and also 7 square root 80
now which perfect square goes into
ninety eight forty nine goes into ninety
eight and you could write ninety eight
as being 49 multiplied by two now what
is the square root of 49 we know the
square root of 49 is 7 and now we need
to multiply 4 by 7 which is 28 so the
final answer for that problem is 28
square root 2
now what about for the next one what
perfect square goes into 80 80 is
divisible by 16 if you take 80 and
divide it by 16 you're going to get 5 so
you can write 80 as being 16 times 5 and
the square root of 16 is 4 so now we
have 7 times 4 which is 28 and so this
is going to give us 28 square root 5 and
so that's it for that problem now what
would you do if you have a problem that
looks like this 3 square root 18 plus 5
times the square root of 72 minus 4
times the square root of 32
how would you simplify this expression
now it's important to understand that at
this point we cannot add the
coefficients of the radicals because
right now what's inside the square root
are different but if they were the same
we could for instance to illustrate that
we can't say 3x plus 5y is 8 X Y that
doesn't work however we can add like
terms
so we could say that 3x plus 5x is a tax
so the only way we can add the
coefficients is if the radicals are the
same for example if I had 4 square root
3 plus 5 square root 3 because the
radicals are the same I can add the
coefficients 4 plus 5 adds up to 9 so
what I need to do in this problem is
simplify the square roots in such a way
that all of these numbers inside the
square root that remains will be the
same so 18 we can write that as 9 times
2 the perfect square that goes into 72
is 36 72 is 36 multiplied by 2 and 32 we
can write that as 16 times 2 now the
square root of 9 is 3 and the square
root of 36 is 6 and the square root of
16 is 4 so now I can multiply 3 times 3
that's gonna give me 9 5 times 6 is 30
and 4 times 4 is 16
so now at this point I can add the
coefficients 9 plus 30 that's 39 minus
16 that's gonna be 23 so the final
answer is 23 square root 2 and so that's
how you could simplify problems like
this you need to simplify each square
root until you get identical square
roots and then you can add the
coefficients so I'm gonna stop it here
today that's it for this video hopefully
you found it useful if you did feel free
to subscribe to this channel thanks
again for watching
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