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Let's look at Square Root Addition and
Subtraction. For example, let's add these
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two numbers together. The first thing
we'll do is simplify each of the square
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root terms by extracting all perfect
squares. In other words, this is equal to
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five the square root of nine five.
45 = nine five and nine is a perfect
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square. And then plus the square root of
four five.
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Twenty = four *five and four is a perfect
square. And now by properties of radicals,*
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this is equal to five the square root of
five plus the square root of four the
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square root of five.
Which is equal to five the square root
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of nine, which is three, the square root
of five, plus the square root of four
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which is two, the square root of five,
which is equal to fifteen times the square
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root of five.
Plus two, the square root of five.
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And finally we can combine these. If we
have fifteen square root of five, plus two
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more, then we have seventeen, square root
of five, which would be our answer. All
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right. Let's look at another example.
Let's simplify this. Again, let's start
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off by simplifying each square root term
by extracting all perfect squares. In
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other words, we can write 27 As nine
three, and nine is a perfect square. And
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then we can write 75, as 25 three, and
25 is a perfect square. And finally we can
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write twelve, as four three, and four is
a perfect square. Again by properties of
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radicals this is equal to the square root
of nine the square root of nine.
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And then + three the square root of 25,
times the square root of three, - two
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the square root of four the square root
of three.
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Which is equal to, the square root of nine
is three, so we have three the square
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root of three plus three the square root
of 25, which is five, the square root of
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three, minus two the square root of
four, which is two.
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Time the square root of three, which is
equal to, three the square root of
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three, and then plus fifteen the square
root of three, and minus four, the
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square root of three.
Again, we can combine these. We have nine
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square root of three, plus fifteen square
root three, which is eighteen square root
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of three - four square root three.
So we're left with fourteen square root of
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three, which would be our answer. Alright,
let's see one more. Again we'll start by
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wor king with each square root separately,
and extracting the perfect squares. In
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other words, this is equal to five the
square root of 24. But 24 is four six,
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and four is a perfect square. Minus four
the square root of 32, but 32 is
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sixteen*2, and sixteen is a perfect
square, plus three * the square root of
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eighteen, and eighteen is nine*2, and nine
is a perfect square. minus two * the
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square root of 54, and 54 is nine six
and nine as a perfect square. Again by
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properties of the radical, this is equal
to five times the square root of four,
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times the square root of six minus four.
the square root of sixteen, the square
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root of two, plus three, the square root
of nine the square root of two - two,
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the square root of nine, the square root
of six, which is equal to, five, the
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square root of four which is two the
square root of six minus four, the
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square root of sixteen, which is four,
the square root of two plus, three the
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square root of nine, which is three, the
square root of two, minus two the square
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root of nine, which is three.
the square root of six. Which is equal
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to, ten, square root of six minus sixteen,
square root of two + nine, square root of
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two, minus six square root of six.
Now it's different here that we didn't see
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in the last two examples, is we are left
with two different types of radicals. We
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have the squared of six in two terms, and
the squared of two in the other two terms.
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So we'll combine terms with common
radical. That is ten square root six - six
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square root.
Would leave us with four square root six.
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And then -sixteen square root two, plus
nine square root two, would be - seven
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square root of two.
Which is our answer.
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This is how we add and subtract with
square roots.
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Thank you, and we'll see you next time.