WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at rational exponents.
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The neat part about rational exponents is you will see that we will develop a way and connect them to our radicals.
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You will see that there are a few rules for working with these rational exponents and a lot of them come for just our rules for exponents.
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We will look at a few ways that you can combine terms that have some these rational exponents on them.
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We have seen many different types of radicals.
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We have seen exponents but there is actually a great connection between the two.
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If you have a radical of some index like a square root or third root and it is raised to a power,
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you can write this in one of the following two ways.
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You can write it as the root of that expression ratio power or you can write it as the expression raised to a fractional power.
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One thing to notice were the location of everything has gone to.
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The power in each of these problems I have marked that off as a that shows up on the top of the fraction.
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The root is going to be the bottom of the fraction.
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You can take any type of radical and end up rewriting it as a fractional exponent or as a rational exponent.
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To get some quick practice with this let us try some examples.
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I have 36^½, -27¹/3, 1³/2 and -9³/2.
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We are going to evaluate these up by turning them into a radicals.
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Okay, so this first one, the way we interpret that is that I'm going to put it under a root with an index of two.
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This is like looking at the √ 36, which is of course just 6.
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For the other one, if I see -27¹/3 that will be like taking -27 to the 3rd root so this one is a -3.
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In the next one notice how we have a power and a root to deal with.
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81³/2
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There are two ways you could look at this.
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You could say this is 81³ and we are going to take the square root or you can take the square root of 81,
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then raise the result to the third power.
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Both of them would be correct, but I suggest going with the one that is a lot easier to evaluate.
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I’m thinking of this one.
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If you take 81 and raise it to the third power we are going to get something very large
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and try to figure out what is the square of that is going to be a little difficult.
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But look at the one on the right, I can figure out what the √ 81 is, I get 9.
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We can go ahead and take 9³ it would be 729.
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One last one, I have 9³/2 and a negative sign out front.
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Let us first write that as the √9³ as for this negative sign it is still going to be out front.
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I have not touched it.
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I'm not including that in everything because there is no parenthesis around the -9 in the original problem.
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I’m starting to simplify this.
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The √9 would be 3 then I will take 3³ and get -27.
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In all of these situations I'm looking at the top of that fraction make it a power, we get the bottom to see what the root needs to be.
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The good part about taking all of our radicals and writing them using these exponents,
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it means that we can use a lot of our tools that we have already developed for exponents and we have done quite a bit of them.
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In fact a quick review, we have a product rule for exponents, a power rule
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and we have gotten different ways that we could go ahead and combine them.
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We also have rules on how to deal with fractions like adding subtractions, subtracting fractions.
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We have our zero exponent rule, our quotient rule and negative exponent rules.
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All of these rules will help us when working with our radicals.
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Just watch on what our base is and what we need to do from there.
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Now using some of these rules if you do have radicals you might be able to combine them under a single root.
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That involves using a common denominator most of the time.
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You can use this tool for working with rational expressions.
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Watch how I find the common denominator for some of my problems and actually get everything under one root.
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Let us do these guys a try.
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64/27⁻⁴/3
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Before I get to that 4/3 part I’m going to apply some of my other rules for exponents and specifically that negative exponent.
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One way I can treat a negative exponent is it will change the location of the things that it is attached to.
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I’m going to write this as (27/64)⁴/3.
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I’m going to give that 4/3 to the top and to the bottom that is using my quotient rule.
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I will end up rewriting the top and bottom using my radicals.
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I’m looking at the 3rd root of 27 and we will take the 4th power of that
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and then we will take the third root of 64 and take the 4th power of that one.
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Let us see what this gives us.
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On top the 3rd root of 27 that will be 3 and 3rd of 64 =4 and both of these are still raised to the 4th power.
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We will go ahead and take care of that multiplication.
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81 / 256
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Being able to take it and write it as radicals, it meshes well with all the rest of our rules.
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Let us use the quotient rule on this next one.
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4⁷/4 ÷ 4⁵/4
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I need to subtract my exponents.
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Good thing both of these have the same denominator this will simply be 4²/4.
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That continues to simplify this would be 4^½ which written as a radical is the √4 which all simplifies down to 2.
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Be very comfortable with switching back and forth between those rational exponents and your radicals.
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On to ones that are a little bit more difficult.
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These will involve trying to simplify much larger expression and some of the terms will have those rational exponents.
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Okay, so here I have (r^¼ y⁵/7)^28 ÷ r⁵.
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I can apply my 28 to both of the parts on the insides since they are being multiplied.
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Let us see what this looks like.
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28 × 1/4 = 28/4 and I have y⁵ × 28 ÷ 7 and all of that is being divided by r⁵.
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Let us see if we can simplify some of those fractions.
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How may times this 4 go into 28? 7 times and 7 will go into 28 four times.
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This is (r⁷ y⁴)/r⁵
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We can go ahead and reduce our y.
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5 of them on the bottom and with 5 of them on top that will leave us with an r² and y⁴.
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Let us try another one.
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This one has a lot of fractions and a lot of negative signs.
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(P⁻¹/5 q⁻⁵/2)/(4⁻¹ p⁻² q⁻¹/5)⁻²
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I’m going to use my rule to apply this -2 to all of my exponents.
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I’m sure that will help get rid of a bunch of different negatives.
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-2 × -1/5 = 2/5, -2 × -5/2 = q⁵
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Then on to the bottom, -1 × -2 = 2 and p⁴ = 4, q²/5.
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That does simplify it quite a bit.
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At least I can see that this guy right here will be 16 but we will also have to reduce these a little bit more.
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I'm going to take care of these ones I want to think what is 2/5 – 4.
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If we can find a common denominator it will helps out with that ones.
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2/5 - 20/5 we will call that one -18/5, so I know that I will have 18/5 on the bottom.
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Let us try it out.
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I have a 16 on the bottom and now I discovered I have a p2^18/5 on the bottom as well.
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These ones we can reduce.
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I want think of 5-2/5.
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The common denominator there will be 5, 25/5 - 2/5 = 23/5 we will put that on top q^23/5.
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We have our final simplified expression.
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In this next two I have some radicals and we will go ahead and write them as our rational and see what we can do from there.
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The top, this would be y²/3 ÷ y²/5
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If I’m going to end up simplifying using our quotient rule, we will look at this as 2/3 – 2/5.
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We need a common denominator on those fractions to put them together.
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Let us look at this as over 15.
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10/15 – 6/15
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This would give us y⁴/15.
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Then we can continue writing this as a radical if I want it to be the 15th root y⁴.
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That is what I was talking about earlier about being able to combine these radicals into a single radical.
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Let us try this other one.
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I have z and I’m looking for the 5th root of it, I will write that as z¹/5.
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When I’m taking the 3rd root of all of that, that is like to the 1/3.
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My rule for combining exponents in this way says I need to multiply the two together.
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Z¹/15
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If I will write this one as a radical them I’m looking at the 15th root of z.
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There are many of the different examples that we can get more familiar with using these radicals and rational exponents.
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Let us do one where we work on writing it using these radicals.
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I’m looking at the 8th root of the entire 6z⁵ – 7th root 5m⁴
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Be careful when it comes to trying to combine things any further from there.
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We have not covered yet on what to do with addition.
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Most of our rules cover our rules for exponents you got to be careful on what you do with addition.
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I’m going to leave this one just as it is and not work on combining.
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One problem that I will have is that my bases are not the same in anyway.
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This is good as it is.
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Be very more familiar with taking the radicals and turning them into rational exponents.
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Remember that the top will represent the power and the bottom of those fractions will represent the index of the radical.
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