WEBVTT



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The other option is let's kind of check this out.

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49 is the same thing as seven squared, right?

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Okay, and then using our laws of exponents, a base to a power raised to a power, we

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multiply

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them, and it's 7.

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Let's try another one for fun.

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I could write 25 as what? 5 squared, which is 5 to the 2 times one half, which is 5.

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Okay, which is the same thing as the square root of 25.

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Uh, let's try a different one.

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What number squared is 30?

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It's not a perfect square, but there is a number, when you square it is 30 and it

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is.

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It's the square root of 30. That's the definition of a number that when you square

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it equals

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that number. The definition is the square root.

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And that is 2 times 1 half, which is the square root of 30 to the power of 1, which

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is the

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square root of 30.

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Okay, so x to the 1 half power is exactly the same thing as the square root of x.

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That's the one-half power.

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What about a one-third power?

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Why is that?

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What number cubed is 8?

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Two.

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Oops.

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Which is two to the three times one third.

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Which is two to the first, which is two.

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Okay.

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which is the cube root of 8. The number that when you cube it is 8 is the cube root

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of 8, which is 2.

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What about a number that doesn't... let's do this... 10 to the one third.

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now. The number that when you cube it is 10 is by definition the cube root of 10.

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And this becomes the cube root of 10. So X to the 1 third is exactly the cube

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root of x. Is that going to work with every single denominator? One fourth? One

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fifth?

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One sixth? Yes. Okay, like,

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81 to the one-fourth.

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What to the power of four is 81?

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Three.

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which is the same thing as the fourth root of 81.

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X to the power of one over any number b is the bth root of that number.

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Uh oh, now it's not just one over a number in the next one.

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I have 64 to the two-thirds power.

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Is this possible yeah, how are we gonna do this or deal with this?

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Okay, so one of our laws

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2 thirds is the same thing as 1 third times 2.

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Well, that's not a law; that's just multiplication.

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One of our laws is x to the a to the power of b is equal to x to the a times b.

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So here I got this part right here.

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So I'm going to write this as 64 to the one third to the power of 2.

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We know the one third power is of 64 is 4.

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And we have 4 squared is 16.

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And we could have done it this way as well.

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64 to the 2 times 1 third, which is 64 squared to the 1 third.

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What is 64 squared?

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Anybody have a calculator?

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It said 4,096.

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Oh yeah.

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Somebody do it on the calculator.

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I think it's 4,096.

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I got it.

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4,096.

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And then what is the cube root of 4,096?

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Yeah, it better be exactly 16 where math is dead if that's not the case

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Okay notice it's a lot easier to first do the root

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Because it makes a small number and then we do the power a

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Lot easier than making the number a lot bigger and then who knows what the cube root

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of that number is

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OK, that gets difficult.

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So in general, x to the a over b can be written two different ways.

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It could be the b-th root of x, raised to the power of a.

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Or the b-th root of x to the power of a.

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Let's try this one.

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So that's going to be the what root?

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Fifth root.

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Whatever the denominator is, is the root.

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To the power of three.

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What is the fifth root of 32?

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Two.

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Two cubed is 8. The other way to do it is to take our base, which is

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32 and just start factoring it.

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2 to the 4, 2 to the 2.

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So that's 2 to the 5th.

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Just

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keep it as exponents, don't mess with anything. A base with an exponent raised to an

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exponent

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that's 2 to the 5 times 3 fifths and what is 5 times 3 fifths 2 cubed which is 8 so

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you leave it all as exponents once you factor the whatever the base is and then

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to just use the rules of exponents to multiply them.

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The square root of 49 is?

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Seven.

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Seven.

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Is it also negative 7?

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OK, this is very important.

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Let's talk about a few things.

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First up, is there a, what's the root here?

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2.

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Or the, OK, that's called the index, or the indices index.

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Tell you the root.

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Square root obviously has an invisible 2 there.

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This is called the radical symbol.

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Radical sign. Okay.

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And the thing inside it.

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It's called the radicand.

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Okay. Whenever you see just the radical symbol,

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that is only the positive square root.

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This is only 7. 49 does have two square roots, positive and negative 7.

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But if you see the radical sign, this is only the positive.

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What is this equal to?

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And sometimes we just say rad 24. This really means the square root

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of 24. We often shorten it and say rad. Is 24 a perfect square? No. Between what

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two integers is this oh definitely between four and five it's gonna be four

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point something

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The only way to write exactly the square root of 24

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is to just leave it like this.

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However, we can simplify this.

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And here's what we do.

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What perfect square goes into 24?

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2 is not a perfect square.

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4 is a perfect square. So this factors to 4 times 6. We agree? Okay. And then one of

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our very important rules, the square root of a times b is equal to the square root

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of

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a times square root of b. Why is that true? Or is that true?

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Let's look at this. Square root of a times b is the same thing as a times b to what

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power?

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One half. And when we have a product raised to an exponent, what can I do with this?

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Raise each factor, a and b, to the one-half power: a to the one half times b to the

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one half.

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Which is one of the laws of exponents we talked about Tuesday, which is the square

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root of a times the square root of b.

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OK, so that's why this guy is true.

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So we're going to use that.

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The square root of 4 times 6 is the square root of 4 times the square root of 6.

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And the square root of 4 is 2.

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OK, so even though this cannot be written as a whole number or a decimal or

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anything,

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We still simplify it to this

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Okay

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So 50 is not a perfect square, but can we simplify it?

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What perfect square goes into 50?

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25 times 2, which I could write as rad 25 times rad 2, which is 5 rad 2.

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Five times square root of 2.

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There's a slightly easier way than trying to figure out what perfect square goes

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into

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it.

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You just take this radicand and you just start factoring it to its prime

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factorization.

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Anytime you have a pair here, it's going to be a perfect square.

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So I could write this as...

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5 squared times 2, which we usually don't write this.

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The square root of 5 squared is 5.

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So typically what I do is I factor it.

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I circle the factor pairs.

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And I know it's going to be 5, rad, whatever is left over.

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2.

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And you're done.

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it's not obvious what the perfect square goes into this

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or it might be to you, but just start factoring it. 72 is what?

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2 times 36. 2 times

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Two times

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9. 3 times 3

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Every pair is a perfect square.

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So it's going to be 2 times 3 times whatever is left over.

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Rad 2.

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So it's 6 rad 2.

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Try this one on your own.

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So, it's going to be 2 times 2 times rad 2 times 3, 4 rad 6.

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If you want to rewrite some people like this, 2 times 2 times 2 times 2 times 2

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times 3,

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Every pair is a perfect square. This becomes 2 times 2 times the square root of 2

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times 3.

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Or you can just do that part in your head.

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Is 40 a perfect cube?

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Instead of thinking what perfect squares go into this sorry perfect cubes just start

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factoring

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Two.

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Two, two.

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10 to

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5

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Now are we looking for pairs?

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Yeah, we need three of a kind because when we have three they multiply to be a

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perfect

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cube.

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So now this is going to be two times the cube root of five.

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You would not say two rad five because rad typically just means square root.

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2 times the cube root of 5.

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Just start factoring: 2 times 80; 2 times 40;

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2 times 20; 2 times 10; 2 times 5.

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Now how many do we need?

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4 We need 4 of a kind

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Which is 16 So it's going to be 2 times the 4th root of

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2 times 5.

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We're going to assume all the variables are positive.

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Now we got numbers and variables.

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Uh oh, what are we going to do there?

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Let's deal with them separately.

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So on the side, let's factor the 20.

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We've got a pair there.

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And if you factor x cubed, it's x times x times x.

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We've got a pair here.

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That's x squared.

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So, on the outside it's going to be 2 times x, and then whatever is left over, which

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is

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a 5 and an x.

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So it's 2x times rad 5x.

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Let's try another one.

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So how many x's on the outside?

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So it's going to be 3x to the power of 3 times the square root of 2x.

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Now we're going to simplify not the square roots, but cube roots.

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So, how many are we looking for at a time?

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Three of a kind.

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So we definitely have a 3 on the outside. How many x's are we looking for?

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3. So we got an x. And then we're left over with a 3 and an x squared.

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That would be it.

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Is this making sense?

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Yes.

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Okay.

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okay we're gonna write out 43 X's can circle every three of them no we need a

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So yeah, we're going to have exactly 14 x cubes

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And a remainder of one more

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So it's going to be X to the 14 times the cube root of X

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Another option is to do this

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What was the X to the 43

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Let's just change this to X to the 43 to the 1 3rd

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Which is X to the 43 times 1 3rd

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Which mixed numbers are terrible, but

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3 goes in the 43 14 times

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14 and 1 third

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But what does fourteen and one third mean?

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That means 14 plus one third.

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Mixed numbers are horrible.

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In algebra, when there is no symbol between factors, it means multiplication.

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Mixed numbers really mean addition.

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Okay, which is x to the 14th times x to the one-third.

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Which is x to the 14th times the cube root of x.

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Same idea of just dividing the exponent by the index and whatever the remainder is,

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that's

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how many are up here.

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Okay, don't start factoring 7,000.

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Is there a perfect power of 4 that goes in 7,000?

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Sorry, 70,000?

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Four zeros is ten to the power of four.

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So let's rewrite this as seven times ten to the power of four.

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And sometimes we could do well

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Four goes into four how many times

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Once okay, so this is gonna go on the outside well

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Let me do this

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We got seven out here right no we got a ten out here

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We got seven in here

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This is a perfect power of 4.

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So the 4th root of x to the 4th is an x.

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4 goes into 7 how many times?

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Once.

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And how many is left on the inside?

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3.

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4 goes into 10 twice.

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How many is left over?

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2.

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This will be z squared.