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Okay, here's a little video on the rational exponents and radicals to finish up that section
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and review section 0.2.
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Let's see, we've got rational exponents.
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Let's start off with those.
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It's not a start off with radicals.
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So we're going to talk about raising this thing to a rational number.
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Okay, remember the other video?
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What's a rational number?
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It's a ratio of integers.
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So there's a rational number.
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Well, you know what?
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One is a integer, so one half is also a rational number.
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So I could raise this to 2 to the power of 5, 7, or 2, the power of 1, half, whatever.
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It doesn't make any difference like that.
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So what does that mean?
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Well, not much here because I can't really do any simplification with this whatsoever.
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But let me talk about something like this.
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What does that mean?
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Okay.
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Well, it turns out that a rational exponent has a meaning in terms of roots and powers.
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The numerator of the rational exponent refers to a power, just like we've been doing in
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the other video.
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But this guy refers to the root that's the number which when multiplied by that many times
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gives us 4.
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Okay.
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So this is really just a tricky way of writing the square root of 4, which happens to be
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2.
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Okay.
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So the square root of 4, and we should just take a moment to talk about how to think
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about these things.
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The square root of 4 is that number which when squared is the radicand in this case
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the number 4.
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Okay.
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And I talk about the square root of 4, it's that number which when squared is 4.
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Well you know what?
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In this case, when we take the radical, we can answer that.
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We can say 2.
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We can say the answer is 2.
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But when we look at a problem like this one right here, unfortunately there is no answer
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to that question.
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The square root of 2 is plainly the square root of 2.
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In other words, remember the square root of 2 is an irrational number and that means it
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can't be written as a ratio of 2 integers.
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In fact, if you know some more about irrational numbers, if we try to represent them as decimals,
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we're going to get a non-repeating decimal that goes on forever.
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So in other words, there isn't any other way to write this number.
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I don't know, I can't write down what number it is when squared equals 2.
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In fact, this is the only way I can write it down.
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So we have this weird thing where people think you can take the square root of 2.
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Well you can't.
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It is the square root of 2.
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The square root of 2 is right here.
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It just turns out that any radicand, which is a perfect square, can be represented by
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an actual integer.
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So in other words, the square root of 4 really isn't a rational number.
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It's an irrational number.
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It's a rational number.
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In fact, it's not only a rational number, but it's an integer, or in this case, it's
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even a natural or counting number.
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So a typical trick question is, what are the irrational numbers here?
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And so people would do things like, they'll write this, they'll write this, they'll write
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this, they'll write this, and they'll say, what are the irrational numbers?
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And so everybody goes, oh, that one.
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Everything that's a radical is an irrational number.
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And so they'll choose this one, this one, this transcendental number, and unfortunately
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they'll be wrong because this is not irrational.
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It's actually a secret code for the number 4, which is a rational number.
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So the answer to this question is, only these two guys are irrational right here.
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Okay, so a little bit on how the rational exponent gets converted into a radical and how
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some radicals are not irrational.
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They're rational.
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One's that are perfect squares, for example, or what?
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Perfect cubes.
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Perfect cubes.
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We're a perfect cube, so let's see.
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So the 27 raised to the 1 third, we were right as what?
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The cube root of 27.
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Now we're going to answer the question, what number when cube is 27?
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And we can answer that question with a 3, so therefore the cube root of 27 is not irrational.
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It is a rational number 3.
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Unfortunately, if I take the cube root of something like, excuse me, I have a square root
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of, square root of 27.
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Hmm, square root of 27.
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Now you can't answer the question, what number, which one's square, is 27?
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And the answer is there is no rational number.
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I can answer that question, so therefore this is an irrational number.
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Okay, so let's do this a little bit more complicated.
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This radical, or excuse me, rational exponent.
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Let's talk about something like this.
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Okay, what if our rational exponent is looks like this?
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Well, first of all, I can turn this into an alternate version because that's 4 times
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1 third.
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And then what, and then that's the same as 4 to the third raised to the fourth or
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what?
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4 to the fourth raised to the third.
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So I can think about it in multiple different ways.
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This I can write as the cube root of 4 raised to the fourth or I could think about that
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being equivalent to, but a cube root of 4 raised to the fourth.
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Okay, so just different ways of saying the same thing.
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Okay, it's nothing tricky, there's nothing to understand really.
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It's just different notations for the same thing.
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So how can you do these a little more complicated here?
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How about if I just take a look at a couple examples to finish this guy out here?
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Which is really well.
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So what am I looking at?
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Oh, I don't know.
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How about the square root 16, 9?
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Well if we go back to this idea that the square root is the rational exponent of the half,
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we know from what we know of exponents that that's really 16 to the half, over 9 to the half.
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And then we can rewrite that as the square root of 16 or the square root of 9.
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And then those are answerable.
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We can answer these.
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What number?
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Which one squared is 16?
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And answer is 4.
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And the same goes for 9.
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It's 3.
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And so we get down 6 to 4 to 3.
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But of course, we could think of this stuff, but we would probably just go right to the
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answer.
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We take the square root of 16 and we'll square root of 9.
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So if we have something like this, you know, 13 over 7, we could easily write that 13 over
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7, but we can't get any further than that.
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So maybe we don't even want to go there because maybe this really isn't any simpler than
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that.
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So we can't answer the question.
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So those are two irrational numbers.
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That's a ratio of irrational numbers.
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So that's what we can do.
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You can.
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So what happens when you add things together?
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Ooh, here's something that is not so good.
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This is not like this.
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That would be a really bad idea to do that.
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It's just confirmed that this is a bad idea.
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What is 16 plus 9?
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That's 25.
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What's the square root of 9?
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3.
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What's the square root of 16?
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4.
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What's the square root of 25?
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That number, which one square root is 25?
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That's 25.
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So what we're saying is 5 is equal to 7.
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It's not true by any means.
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So don't be going down that road.
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There is no rule about this, just like there was never a rule about anything like this.
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There was no rule that said that was 3 squared plus 2 squared by gosh no.
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So remember, this is a radical act, irrational exponent.
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It's the same as raising it to the half.
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So it's no different than this.
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This isn't true.
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So this isn't going to be true either.
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Only when they're multiplied, not when they're two terms, this is going to work.
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So let's try that.
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Let's try that.
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What is that going to be?
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So what about the square root of 9 times 16?
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What's that going to be?
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Well, that's the same thing as 9 times 16 and a half.
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And then we know from exponents, that's the same as this.
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And we also can take that, put it back like that.
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So we could have multiplied 9 times 16 and done that.
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But that's just going a long way.
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It's making a mess out of it when we really need to define all the perfect squares that
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were in that number, all the perfect vector squares are the ones that are going to have
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that answerable question.
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So if you don't have a perfect vector square, then you're not going anywhere.
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That's it.
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You're done.
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You're irrational.
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OK, remember, if you're not a perfect square, you must be irrational.
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No, that doesn't quite work that way.
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But anyway.
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Let's see.
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So how are we going to finish this up a little bit more here about another problem that's
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in about this one?
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I got that 8 times 64.
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And I'm taking the cube root of that.
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OK, well, I could multiply 8 times 64 and by gosh.
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But that's a mess.
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So why don't I just take the cube root of 8 times the cube root of 64?
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And let's see if they can answer that question.
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Is there some number which one cube is 8?
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The answer is 2.
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Is there some number which one cube is 64?
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Well, you probably don't know that one offhand, but it turns out to be 4.
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OK, so the answer is 8.
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So that's the way it works, some simplification like that.
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Let's see.
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How about if we take a look at, well, they've got something like this.
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OK, let's take care of the radicand.
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Let's get as simple as we can before we decide whether there's any perfect squares in
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the room.
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So there's still a 4.
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And this two of those factors and one downstairs leaves me one downstairs.
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And three up and two there.
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So that leaves me two down there and still the square root.
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So for each factor, I can talk about whether there is a perfect square.
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What's the perfect square?
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What number squared is 4?
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2.
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What number squared is y squared?
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Well, y squared is y squared.
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So it's a y here.
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And what about that?
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Sorry.
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That's a no-go.
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There's only the square root of x down there.
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And there is no number, which one squared is x, at least not algebraically.
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So we also got another issue which they don't like, you know, leave, which is supposed to
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be a rationalized denominator.
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So the denominator is currently looking irrational.
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So there, change that over.
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That's another thing you would have learned in your main algebra.
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So how do you do that?
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You multiply by number, which one squared is the radicand x, right?
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Square root of x times square root of x.
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That's the number, which one squared is x.
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So but I've got to multiply by the magic number one, right?
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When I flip something over something, it's one.
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If I multiply by one, I multiply by the identity element, and I haven't changed what
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it's here.
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So that's the tricky version of one.
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So we're going to get to my square root of x divided by what?
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PlaneOnex.
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And I guess I'm done.
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Okay.
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So, well, that should be enough.
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I don't think this is super critical stuff going forward, but a little practice with this
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is a good way to get warmed up for thinking out.