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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â each square root term by extracting all perfect squares.

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

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

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

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

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

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

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

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

Â 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

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

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

Â three and minus four times the square root of three.

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

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

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

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

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

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

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

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

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

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

Â 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â seven square root of two, which is our answer.

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

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

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