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[SOUND] Let's look at square root, additions and subtraction.

[SOUND] For example, let's add these two numbers together.

The first things we'll do is we'll simplify each of the square root terms by

extracting all perfect squares. In other words, this is equal to five

times the square root of nine times five. 45 = 9 * 5 and 9 is a perfect square,

and then, plus the square root of four times five,

20 4 * 5 and 4 is a perfect square. And now, by properties of radicals, this

is equal to, five times the square root of nine times the square root of five

plus the square root of four times the square root of five,

Which is equal to five times the square root of nine, which is three, times the

square root of five plus the square root of four, which is two, times the square

root of five, which is equal to 15 times the square root of five plus two times

square root of five. And finally, we can combine these.

If we have 15 square root of five plus two more, then we have 17 square root of

five, which would be our answer. All right, let's look at another example.

[SOUND] Let's simplify this. Again, let's start off by simplifying

each square root term by extracting all perfect squares.

In other words, we can write 27 as 9 * 3 and nine is a perfect square.

And then we can write 75 as 25 * 3 and 25 is a perfect square.

And finally we can write 12 as 4 * 3 and four is a perfect square.

Again, by properties of radicals, this is equal to the square root of nine times

the square root of three, and then plus three times the square root of 25 times

the square root of three minus two times the square root of four times the square

root of three, which is equal to the square root of nine

is three. So we have three times the square root of

three plus three times the square root of 25, which is five, times the square root

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of three minus two times the square root of four, which is two, times the square

root of three, which is equal to three times the square

root of three. And then plus 15 times the square root of

three and minus four times the square root of three.

Again, we can combine these. We have three square root of three plus

15 square root three which is 18 square root of three minus four square root

three. So we're left with 14 square root of three, which would be our answer.

Alright, let's see one more. [SOUND] Again,

we'll start by working with each square root separately and extracting the

perfect squares. In other words, this is equal to five

times the square root of 24, but 24 is 4 * 6 and four is a perfect square, minus

four times the square root of 32, but 32 is 16 * 2 and 16 is a perfect square,

plus three times the square root of 18 and 18 is 9 * 2,

and 9 is a perfect square, minus two times the square root of 54)

and 54 is 9 * 6, and nine is a perfect square.

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Again, by properties of the radical, this is equal to five times the square root of

four times the square root of six minus four times the square root of 16 times

the square root of two plus three times the square root of nine times the square

root of two minus two times the square root of nine times the square root of 6.

Which is equal to five times the square root of four, which is two times the

square root of six minus four times the square root of 16, which is four times

the square root of two plus three times the square root of nine, which is three

times the square root of two minus the square root of nine, which is three times

the square root of six, which is equal to 10 square root of six minus 16 square

root of two plus nine square root of two minus six square root of six.

Now, what's different here that we didn't see in the last two examples, is we're

left with two different types of radicals. We have the square root of six

in two terms and the square root of two in the other two terms.

So, we'll combine terms with common radicals, that is ten square root of six

minus six square root of six would leave us with four square root of six.

And then, -16 square root of two plus nine square root two, would be negative

seven square root of two, which is our answer.

And this is how we add and subtract the square roots.

Thank you and we'll see you next time. [SOUND]