Let's look at multiplication involving square roots. For example, let's simplify
this expression here. Let's begin by first simplifying these square roots seperately.
So, what is the square root of twenty? Well, let's extract all perfect squares
here. In other words, twenty is four five.
And then by properties of radicals, this is equal to the square root of four times
the square root of five, which is equal to two times the square root of five.
And what about the square root of 80? This is equal to the square root of sixteen
five, which is equal to, again, by properties of radicals, square root of
sixteen times square root of five or four times square root of five.
So, let's put these simplifications back into here which gives us, that this is
equal to four times two square root of five because we just found that the square
root of twenty was two square root of five and then times the square root of 80. And
the square root of 80, we just found to be four square root of five.
And now by commutativity, we can regroup this multiplication as four two four
and then, times the square root of five times the square root of five, which is
equal to four times two is eight times four is 32. And then, square root of five
times square root of five is just five. Because remember, we have the following
property that the square root of a rimes the square root of b ids equal to the
square root of a b, for a and b non-negative. So, the square root of five
times the square root five would be square root 25 or five.
And this is equal to 32 five is 160, which would be our answer. Now, we could
have began a bit different and just applied this property from the very
beginning. In other words, this is equal to four times the square root of twenty
times 80, which is equal to four times the square root of 1600. And if we simplify
this, we'll get to the same answer. But 1600 is a pretty big number so we decided
to simplify the smaller radicals first. Alright. Let's look at another example.
Let's simplify this. What we'll do is we'll begin by foiling. That is, this is
equal to five square root of three three square root of three + five square root of
three -two square root of two + four square root of two three square root of
three and finally, plus four square root of two -two square root of two.
Which is equal to five times three is fifteen, nine times the square of three
times three or nine minus ten times the square root of six + twelve times the
square root of six and -eight times the square root of four, which is equal to
fifteen three, the square root of nine is three, then -ten square root of six +
twelve square root of six - eight the square root of four which is two, which
gives us 45 - ten square root of six + twelve square root six - sixteen.
And now, we'll combine these numbers here as well as the terms with the square root
of six in them, which gives us 45 - sixteen is 29. And then, - ten square root
six plus twelve square root six is positive two square root six, 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.