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