factor
## How To Simplify Square Roots

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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