Let's look at Square Root Addition and Subtraction. For example, let's add these
two numbers together. The first thing we'll do is simplify each of the square
root terms by extracting all perfect squares. In other words, this is equal to
five the square root of nine five. 45 = nine five and nine is a perfect
square. And then plus the square root of four five.
Twenty = four five and four is a perfect square. And now by properties of radicals,
this is equal to five the square root of five plus the square root of four the
square root of five. Which is equal to five the square root
of nine, which is three, the square root of five, plus the square root of four
which is two, the square root of five, which is equal to fifteen times the square
root of five. Plus two, the square root of five.
And finally we can combine these. If we have fifteen square root of five, plus two
more, then we have seventeen, square root of five, which would be our answer. All
right. Let's look at another example. Let's simplify this. Again, let's start
off by simplifying each square root term by extracting all perfect squares. In
other words, we can write 27 As nine three, and nine is a perfect square. And
then we can write 75, as 25 three, and 25 is a perfect square. And finally we can
write twelve, as four three, and four is a perfect square. Again by properties of
radicals this is equal to the square root of nine the square root of nine.
And then + three the square root of 25, times the square root of three, - two
the square root of four the square root of three.
Which is equal to, the square root of nine is three, so we have three the square
root of three plus three the square root of 25, which is five, the square root of
three, minus two the square root of four, which is two.
Time the square root of three, which is equal to, three the square root of
three, and then plus fifteen the square root of three, and minus four, the
square root of three. Again, we can combine these. We have nine
square root of three, plus fifteen square root three, which is eighteen square root
of three - four square root three. So we're left with fourteen square root of
three, which would be our answer. Alright, let's see one more. Again we'll start by
wor king with each square root separately, and extracting the perfect squares. In
other words, this is equal to five the square root of 24. But 24 is four six,
and four is a perfect square. Minus four the square root of 32, but 32 is
sixteen2, and sixteen is a perfect square, plus three the square root of
eighteen, and eighteen is nine2, and nine is a perfect square. minus two the
square root of 54, and 54 is nine six and nine as a perfect square. Again by
properties of the radical, this is equal to five times the square root of four,
times the square root of six minus four. the square root of sixteen, the square
root of two, plus three, the square root of nine the square root of two - two,
the square root of nine, the square root of six, which is equal to, five, the
square root of four which is two the square root of six minus four, the
square root of sixteen, which is four, the square root of two plus, three the
square root of nine, which is three, the square root of two, minus two the square
root of nine, which is three. the square root of six. Which is equal
to, ten, square root of six minus sixteen, square root of two + nine, square root of
two, minus six square root of six. Now it's different here that we didn't see
in the last two examples, is we are left with two different types of radicals. We
have the squared of six in two terms, and the squared of two in the other two terms.
So we'll combine terms with common radical. That is ten square root six - six
square root. Would leave us with four square root six.
And then -sixteen square root two, plus nine square root two, would be - seven
square root of two. Which is our answer.
This is how we add and subtract with square roots.
Thank you, and we'll see you next time.