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# 更多簡化根式 (英): 更多簡化根式

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- In this video, I'm going to do some more examples of
- simplifying radical expressions.
- But these are going to involve adding and subtracting
- different radical expressions.
- And I think it's a good tool to have in your toolkit in
- case you've never seen it before.
- So let's do a few of these.
- So let's say I have 3 times the square root of 8-- we
- learned before that's actually the principal square root of
- 8, or the positive square root of 8-- minus 6 times the
- principal square root of 32.
- So let's see what we can do to simplify this.
- So first of all, 8, we can write that as 2 times 4.
- When 4 is a perfect square, you might
- already recognize that.
- We could further factor that into 2 times 2.
- But I don't think we need to.
- So we can rewrite 3 square root of 8 as 3 times the
- square root of 4 times the square root of 2.
- This is the same thing as the square root of 4 times 2,
- which is the square root of 8.
- So this term is the same thing as that term.
- And then, let's look at 32.
- We want to do the square root of 32.
- 32 is 2 times 16.
- Once again, 16's a perfect square, so
- we could stop there.
- If you didn't realize that, you would factor
- that as 4 times 4.
- You'd see that twice.
- You could even go even further down to 2 times 2 and all of
- that, but you see immediately that's a perfect square, so we
- can stop there.
- So this second expression can be written as minus 6 times
- the square root of 16 times the square root of 2.
- This right here-- I want to be clear-- is the same thing as
- the square root of 16 times 2.
- You can separate out.
- The square root of 16 times 2 is the square root of 16 times
- the square root of 2.
- We saw that with our exponent properties.
- Now, what does this first term simplify to?
- This is 3 clearly.
- This right here is a 2.
- So you have 3 times 2 times the square root of 2.
- That is 6 times the principal root of 2.
- And then from that we're going to subtract-- well, what's
- this term right here?
- That is positive 4.
- So 6 times 4 is 24 times the square root of 2.
- And we're not done yet.
- If I have 6 of something and I'm going to subtract from
- that 24 of that same something, what do I have?
- I have 6 square roots of 2 and I'm going to subtract from
- that 24 square roots of 2, well, this is going to be
- equal to 6 minus 24 is negative 18 square roots of 2.
- And hopefully, this doesn't confuse you.
- Remember, if we had 6x minus 24x, we would have minus 18x
- or negative 18x.
- Now, instead of an x, we just have a square root of 2.
- 6 of something minus 24 of something will get us negative
- 18 of that something.
- Let's do another one.
- Let's say I have the square root of 180 plus 6 times the
- square root of 405.
- So this is really an exercise in being able to simplify
- these radicals, which we've done before.
- But you can never get too much practice doing that.
- So let's just do the factorization
- of these right here.
- So 180 is 2 times 90, which is 2 times 45,
- which is 5 times 9.
- And we can factor 9 down more into 3 times 3 to realize it's
- a perfect square, but we could leave it like that.
- So this first term right here we can write as the square
- root of 2 times 2 times the square root of 5 times the
- square root of 9.
- I'm going to put the square root of 9 out front.
- So square root of 2 times 2 times the square root of 5
- times the square root of 9.
- Now, what is this second term equal to?
- So let's factor it out.
- 405.
- That is 5 times-- I think it's 81.
- But just to verify, 405, 5 doesn't go into 4, so
- let's go into 40.
- 5 goes into 40 eight times.
- 8 times 5 is 40.
- Subtract.
- You get a 0.
- Bring down the 5.
- 5 goes into 5 one time.
- Right, 81 times.
- 81 is 9 times 9.
- You could factor more if we were trying to do the fourth
- root or something like that, but we want to just do a
- square root.
- We have a 9 and a 9, so no need to factor any more.
- So this second expression right here is plus 6 times the
- square root of 9 times 9 times the square root of 5.
- So what is this?
- This is 3.
- This is 2.
- This is the square root of 4.
- So it's 3 times 2 is 6.
- So we have 6 square roots of 5.
- Plus-- what's this right here?
- The square root of 9 times 9, the square root of 81.
- That's, of course, just 9.
- So 6 times 9 is 54, so plus 54 square roots of 5.
- And then, what do we have left?
- We have 6 of something plus 54 of something.
- That's going to be equal to 60 of that
- something just like that.
- Let's just do one more and we're going to have some
- abstract quantities here.
- We're going to deal with some variables.
- But I really just want to do it to show you that the
- variables don't change anything.
- Let's say if we have the square root or the principal
- root of 48a.
- And I'm going to add that to the square root of 27a.
- So once again, let's just factor the 48 part.
- We'll leave the a aside.
- So 48 is 2 times 24, which is 2 times 12.
- Sorry, 2 times 12, which is 3 times 4.
- So we could rewrite this first expression here as the square
- root of 2 times 2 times the square root of 4 times the
- square root of 3.
- Now, you might have done it a quicker way.
- You might have just factored into 3 and 16 and immediately
- realized that 16 is a perfect square.
- But I did it just kind of the brute force way.
- You'd get the same answer either way.
- And, of course, not just the square root of 3, you also
- have the square root of a there.
- So I'll just put the a right over here.
- I could put it in a separate square root, but both of these
- aren't perfect squares, so I'll leave both of these under
- the radical sign.
- Now, 27 is 3 times 9.
- 9 is a perfect square root, so we can stop there.
- So this second term, we can rewrite it as the square root
- of 9 times the square root of 3a.
- And in both of these you can kind of view it I'm skipping
- an intermediate step.
- The intermediate step, I could have written that first
- expression as the square root of 9 times 3a and then
- gone to this step.
- But I think we have enough practice realizing that 9
- times 3a, all of that to the 1/2 power, or taking the
- principal root of all of that is the same thing as taking
- the principal root of 9 times the principal root of 3a.
- So that's the step I skipped in both of these.
- But hopefully, that doesn't confuse you too much.
- And so, this term right here is going to be a 2.
- This term right here is going to be a 2.
- So this is going to be 4 times the square root of 3a.
- And then this over here, this right here, is a 3.
- So this is going to be plus 3 times the square root of 3a.
- 4 of something plus 3 of something will be equal to 7
- of the something.
- Anyway, hopefully, you found that useful.

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