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Let's take a look at adding or subtracting radical
expressions.
When we add radical expressions, like 7 root 3
plus 5 root 3, what we want to do is combine like radicals.
Now for like radicals, two radicals have to
have the same index.
So the same kind of a root like a square root, a cube
root, a fourth root, and the same radicand, which would be
whatever is on the inside of the root symbol.
So if we look here at 7 root 3 plus 5 root 3, these are like
radicals because the roots are both square root of 3,
square root of 3.
That means we can combine them in exactly the same way you
would combine like terms.
Let's say we've got 7x plus 5x, well that
would give us 12x.
So we combine the coefficients and keep the x.
Here we will combine the coefficients, 7 plus 5 is 12.
And keep the root 3.
And that would be it.
So if we look at the next one, we have 3 radicals.
And we have to first decide which ones, if
any, are like radicals.
This one is the square root of 5.
This one is also square root of 5.
So these are like radicals.
And it doesn't matter what's on the front.
The coefficients can be different.
This one is a 2.
This one has a 1.
Even though it's not showing, implicitly the 1 is there.
So coefficients don't matter, but root
5, root 5 like radicals.
This one's root x.
The inside is different.
Therefore, it's not a like radical.
So we're not going to combine that one with the other two.
So we'll just combine these two.
We combine the coefficients.
1 plus 2 is 3.
Keep the radical root 5.
And just bring this other guy down, minus 8 root x.
And that would be our answer.
And the last one we have four different radicals.
And let's decide which ones, if any, are like radicals.
Here we have square root of x, fourth root x, square root x,
fourth root x.
All right that means that square root x and square root
of x are both like radicals.
And the fourth root of x and the other fourth root of x are
like radicals.
So we'll combine them separately.
First let's take care of the root x.
We have 5 root x minus 2 root x.
So 5 minus 2 is 3.
Keep the root x.
And for the other ones we have, plus 3 fourth root x
plus 6 fourth root x.
3 plus 6 is plus 9.
And we keep the fourth root x.