[SOUND] Let's look at multiplication involving square roots.

[SOUND] For example, let's simplify this expression here.

Let's begin by first simplifying these square roots separately.

So what is the square root of 20? Well, let's extract all perfect squares here.

In other words, 20 is 4 * 5, and then by properties of radicals, this is equal to

the square root of 4 * the square root of 5 which is equal to 2 * the square root

of 5. And what about the square root of 80?

This is equal to square root of 16 * 5 which is equal to again by properties of

radicals, square root of 16 * square root of 5 or 4 * square root of 5.

So let's put these simplifications back into here, which gives us that this is

equal to 4 * 2 square root of 5. As we just found that the square of 20

was 2 square root of 5 and then times the square root of 80.

And the square root of 80, we found to be 4 square root of 5.

And now by commutativity we can regroup this multiplication as 4 * 2 * 4 and then

* √5 * √5, which is equal to 4 * 2 = 8 * 4 = 32.

And then √5 * square root of 5 is just 5. Because remember we have the following

property, that the square root of a * of b = to the square root of a * b for a and

b non-negative. So, the square root of 5 times the square

root of 5 would be the square root of 25, or 5.

And this is equal to 32 * 5 = 160 which would be our answer.

Now we could have begun a bit different and just applied this property from the

very beginning. In other words this is equal to 4 * the

square root of 20 * 80 which is equal to 4 * the square root of 1600.

And if we simplified this, we'd get to the same answer.

But 1600 is a pretty big number, so we decided to simplifiy the smaller radicals

first. All right, lets look at another example.

[SOUND] Lets simplify this. What we'll do, is we'll begin, by

foiling. That is, this is equal to 5 square root

of 3 * 3 square root of 3 + 5 square root of 3 * -2 square root of 2 + 4 square

root of 2 * 3 square root of 3 and finally + 4 square root of 2 * -2 square

root of 2, which is equal to 5 * 3 is 15 and then times the square root of 3 * 3

or 9, - 10 * the square root of 6 + 12 * the square root of 6 and - 8 * the square

root of 4 which is equal to 15 * 3, the square root of 9 is 3.

And then -10 square root 6 + 12 square root 6 - 8 * the square root of 4 which

is 2. Which gives us 45 - 10 square root of 6 +

12 square root 6 - 16. And now, we'll combine these numbers

here, as well as the terms with the square root of 6 in them which gives us,

45 - 16 = 29. And then, -10 square root 6 + 12 square

root 6 is +2 square root 6 which would be our answer.

And this is how we work with multiplication involving square roots.

Thank you, and we'll see you next time.

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