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 integer exponents.
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What does it mean? You take a quantity and raise it to a power.
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A lot of other things we will also be picking up along the way.
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Let us get a good overview.
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We will start off by interpreting what it means to raise something to a power when that power is a nice whole number.
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We will see that we can often combine things using the product and the power rule because all these things have the same base.
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We will see what it means to raise things to a 0 power and get into the quotient rule.
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And get into understanding what happens when you raise something to a negative power.
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Watch for all of these things to play a part.
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One way that we can look at multiplication is that it is a packaging up of addition.
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I have the value of x and we know that I have added together 5 times.
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Rather than trying to string out to x + x + x over and over again I can write that same thing by just saying 5 multiplied by x.
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It is a great way to take a long addition problem and package it all down into a nice single term.
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Exponents are the things that we are interested in as long as we are dealing with whole numbers you can look at that as packaging up some multiplication.
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Here I have those x’s once again, I have 5 of them all being multiplied together but rather writing all that, I'm going to use my exponents.
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Specifically the bottom here will be our base.
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Remember, that is what we are multiplying over and over again.
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The exponent itself tells us how many times.
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Let us get some practice with just understanding how exponents work before moving on.
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With this, I simply want to write them into their exponential potential form.
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If I have 5 × 5 × 5, I can see that it is the 5 being repeatedly multiplied and I did it 3 times.
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I could consider this one like 5³.
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If I wanted to take this one little bit farther, this is equal to 125.
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You have to focus on what is being multiplied over and over and over again to properly identify your base.
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In this one, we have -2 × -2 × -2, we can see that it is being multiplied by itself 4 times.
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We want the entire value of -2 multiplied by it self 4 times so we will use an exponent of 4.
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If we had to simplify this one, I have -2 × -2 × -2 × 2 and that would be 16.
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Keep an eye on the base, so you know what is being multiplied and the exponent will tell you how many times to do that.
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In this next two, we just want to evaluate the exponential expression.
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Let us see if we can identify the base here.
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This is a 3⁴ that happens to be negative sign out front and my next example, it is almost the same thing but it is -3⁴.
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Watch how I treat both of these a little bit differently.
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In this first one, the only thing that is considered the base is actually that number 3, 3 × 3 × 3 × 3 multiplied by itself 4 different times.
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What should I do with that negative sign out front, it is still out front, but since it is not part of the base and it only shows up once.
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I can take care of it from here, 3 × 3 × 3 × 3 a lot of 3’s in there, but that will equal 81.
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Of course that negative sign is out front along for the right.
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In the next one, these parentheses are highlighting that the entire -3 is what is in the base so I'll repeat the entire -3.
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Look at this one, I'm dealing with a -3 multiplied by itself 4 times.
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I know that a negative × negative would be positive and another negative × negative would be a positive,
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the answer to this one action turns out to be 81.
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Use those parentheses to help you know what is in the base and it is good for all of those negatives.
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One of our first rules that we can throw into the mix is the product rule for exponents.
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This will help us when we have two things and both of them are raised to an exponent.
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It is a great way to actually put them together so they only have one thing raised to an exponent.
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The way this rule works is, we want to make sure that both the bases are exactly the same.
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I have a and a as my base, if they are the same then it says we simply need to take each of their exponents and add them together.
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It seems like a little bit of an odd rule how you can always start with multiplication and end up with addition
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but let me show you an example of why this works.
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I’m going to base that off just what you know about exponents that it is repeated multiplication.
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Here I have 2⁴ power be multiplied by 2³, let us pretend I knew nothing about the product rule.
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I can start interpreting this by using repeated multiplication, so 2⁴ will be 2 × 2 × 2 × 2 until I done that 4 times in a row.
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I can do the same for 2³ that would be 2 × 2 × 2.
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You can see what we have here, the whole bunch of 2s are all being multiplied together and there is even no real need to put on those parentheses.
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I will just write 2 × 2 × 2 until I got all of them written down.
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As long as I have all of these 2s written out, it was said earlier that exponents are just repeated multiplication,
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I can actually package these all backup.
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If you count how many are here, it looks like we have 7 total.
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I can say that this is 2⁷.
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Now comes the important part, if you look at our original exponents that we have used the 4 and 3 and you add them together
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you can see that sure enough, it does add to 7.
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The reason why this is happening is because you are simply throwing in a few more numbers to multiply by and incrementing up that exponent.
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That is why the product rule works.
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Let us go ahead and practice with it now.
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In our first one I have -7⁵ × -7³.
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The first thing you want to recognize is both of those bases are the same, so we will just focus on their exponents the 5 and 3.
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This one as -7^(5+3) and then I can go through and add those 2 things together, -7⁸.
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You could simplify it further from there if you want to but I’m just going to leave it like that so you can see the product rule in action.
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The next one will involve a little bit more work and that is why I put in the -4 and 3.
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Before I can get too far, I’m going to start rearranging the location of the 4 and the p's raised to the exponents.
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What allows me to do this is the commutative property for multiplication that the order of multiplication does not matter.
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This is just helping me organize things a little better so that I have my numbers all in one spot and my exponents on one spot as well.
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That looks good.
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Put the -4 and 3, I will just straight multiply those two together and get -12.
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With the other two, I can see that they are both the same base of p.
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I will look at adding their exponents together so p^(5+8).
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Here is -12 p^13 not bad.
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One last, let us see if we can use the product rule in that one.
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Looking at this one you will know that they both have a base of 6.
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One is raised to the 4th power and one is raised to the power of 2.
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But there is a slight problem with this one.
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This one is dealing with addition and the product rule, the one that we are interested in is for multiplication.
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Since the product rule only applies to multiplication, it means we can not use the product rule here.
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Even though that my bases are the same it simply not going to work here.
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If I did want to continue this one out, I would not use the product rule.
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I would just simply take 6⁴ and see what value that is.
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I will get 1296 then I will get 6² = 36 and I will add the two together 1332.
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You can notice how we do not use the product rule since it has addition there on the bottom.
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Let us try a few more examples and see if we can recognize our product rule in action.
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I have thrown in a few more variables this time.
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The first one is n × n⁴.
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It is tempting to say hey maybe this one is just the n⁴, but if you see missing powers like on that first one, assume that they are 1.
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That way you actually have something to add.
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This would be n1+ 4 and now that they actually have your exponents added together, we will get n⁵.
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Now you know if you strictly look at the product rule it only talks about putting two things together, you can even apply it for more than two things.
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In this one, I have z² × z⁵ × z⁶ and we can apply the product rule as long as we do it two at a time.
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Let us just take the first two z’s here.
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This would be z^(2+5), we will get to that other z in a bit.
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I can see that I have a single z^(2+5) and z⁶, that simplifies into z⁷.
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I can put these two together z^(7+6) and that would be z^13.
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We could have even taken a much bigger shortcut if we simply just added all 3 of them together.
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You can definitely do this as long as all of their bases are the same, and you are dealing with multiplication.
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One more, this one is 4² multiplied by 3⁵.
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We are definitely dealing with multiplication but again, this one has a little bit of a problem.
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Notice how this one, the bases are not the same.
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The condition for using that product rule is that we need the base is the same
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and we need multiplication and so we simply can not use the product rule here.
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If we are going to go any farther with this one, we have to evaluate these exponents.
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4² would be 16 and then I will go ahead and multiply that by 3⁵ = 243 and then multiply the two together, so 3888.
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Be very careful that the conditions of the product rule are met before you try and use it.
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Another good rule that we can use for combining things together is known as the power rule.
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The way the power rule works is, we have something raised to an exponent and then we go ahead and re-raise all of that to another exponent.
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It is like we have exponents of exponents.
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The way the power rule handle these is we will take both of the exponents and we will actually multiply them together.
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To convince you that this is the way it should work, we are going to deal with another example that has just numbers in it,
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and show you that this is the way you should combine your exponents in this case.
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We are going to take a look at 4³ and all of that is being squared.
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Say you knew nothing about the power rule, how could you end up interpreting this?
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One way is you could use again repeated multiplication to interpret your exponents.
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In the first exponent I'm going to interpret is this two right here.
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I’m going to take my base, 4³ and it will be multiplied by itself twice, you can see I have two of them.
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I want to go further with this, now I will interpret both of these 3s as 4 being multiplied by itself 3 times.
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Now that I have expanded it out entirely, I’m going to work to go ahead and package it all up.
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What I can see here is that I have a whole bunch of 2s and they are all going to be multiplied together.
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In fact I have a total of 6, this is 4⁶.
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Go ahead and observe exactly what happened with the exponents.
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Originally we started with 3 and 2 and if you multiply those together like the power rule says we should sure enough, you get 6.
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If you fall along the process you can see why that happens.
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Now that we know more about this rule, let us move on.
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The power rule which we learned earlier can be applied to multiplication and division.
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The way it does it is we can take two things that are being raised to a power and then that giving each of them that exponent.
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When dealing with division, if we have things being divided, say, a ÷ b we can take that exponent and give it to each of them.
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What I’m trying to say here is that if you have an exponential expression and it is raised to a power,
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you are going to apply it to all the parts on the inside as long as those parts are being multiplied or divided.
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This one helps us when we get to those larger expressions.
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Let us try a few of these up, we simply want to use the power rule for each expression rated out.
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The first one is 6² and all of that is being raised the fifth power.
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I'm not going to expand that out, I’m just going to go directly to the rule.
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This would be 6 then we have 2 × 5.
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Taking care of just the exponents this would be 6^10.
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I could continue to try and figure out what number that is but I’m only interested in considering what the power rule is.
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Let us try another one.
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This is z^4and all of that is being raised to the 5th.
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I can take my exponents and I will multiply them together.
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This will be z^20.
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Let us get into a few that are a little bit more complicated and see how we can tackle this.
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In this next one I have 3 × a² × b⁴.
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One of the first things I need use is my power rule and give that 5 to all of the parts.
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Let us write that to each of these.
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I’m going to do the 5³, I'm going to give it to the a² and I'm going to give it to the b⁴.
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It must go on to all of those parts and it is okay since all these parts were multiplied in the original.
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Now that I have this, I will go ahead and start to simplify each of these.
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My 3⁵ is 243, then I will multiply these exponents together and get a^10.
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We will multiply the 4 and 5 together and would give us b^20.
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This one is a completely condensed down far as I can go.
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Let us see one that involves division.
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I'm going to give the 3 to the top and bottom.
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The top will be 3³ and the bottom will be x³.
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This will give us 27 all over x³.
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Let us try one more 4 + x².
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What does the power rule say we should do with this one?
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I threw this one in there because you should not use the power rule on this one.
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Why not? Why can not we use the power rule here?
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Notice the parts on the inside, they are being added 4 + x instead of being multiplied.
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Since it is being added instead of multiplied, I can not simply give the 2 to all the parts.
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This situation down here is completely different from this one because of that addition.
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How could I take care of this?
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What we will see in one of our future lessons is that we will handle this one using repeated multiplication
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and we simply have to multiply those two terms together.
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For now I want to point out is you should not use the power rule on that situation simply does not work.
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All of these rules are extremely important and you should be very comfortable with them.
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I recommend memorizing all of them.
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Usually a lot of practice with them will get you comfortable with recognizing when you should and should not use them.
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In addition, you should be comfortable with these that you can use more than one of them in a single problem.
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Maybe you will end up using the power rule and then end up using the product rule all in the same one to condense and simplify an expression.
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That is how comfortable with these rules you should be.
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We can get a better handle and the least a little bit more comfortable with these we will give it a try.
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We will try and use many of these rules together to simplify the following expressions.
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Okay, this first one is (5k³ / 3)².
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One of the very first rules that I'm going to use, is I'm going give that 2 to the numerator and to the denominator of that fraction.
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You got to be careful in doing this, the top is 5k³ and the bottom is just 3.
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We will give 2 to each of those.
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Now that I have taken care of that rule I can see I have a little bit more of work to do on top.
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I have multiple parts in there all being multiplied and I’m going to give the 2 to each of those parts.
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Here is my 5k³ so we will give the 2 to the 5 and we will give the 2 to the k³.
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Now that we spread out our variable amongst all of the parts that are being multiplied and divided, let us see if we can continue.
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5² that would be 25 and now I have k³ and all of that is being raised to the power of 2.
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What rules should we use for that?
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As we only have a single base in there, this is that situation where we multiply the two together so this will be k⁶.
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Now I simply have 9 on the bottom from 3².
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I do not see anything else that I can combine or any other rules that I can use, this is good to go.
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Be careful in using multiple rules.
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Let us take a peek at the second one here.
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This one is (-3 × x × y²)³.
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In addition that is being multiplied by (x² × y)⁴, lots and lots of different exponents flying around.
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Let us recognize that we do have some multiplication on the insides of those parentheses,
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I will be able to spread out that exponent amongst all of its parts.
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What parts do I have in there?
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I have -3, x, y² so we will put 3 on each of those.
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Let us do the same thing for the other one.
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I have x² and y⁴, both of these looks like they need 4.
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(x²)⁴ y⁴, not bad.
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Now that I have applied that rule, let us go through and start cleaning other things up.
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Starting at the very beginning I have -3³.
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That is -3 × -3 × -3 = -27.
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Right now I have x³ just as it is.
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It looks like this next part that is a good place where I can multiply the exponents together, y⁶.
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Here is a couple of more situations where again I can just multiply those exponents together x⁸ y^16.
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It is important not to stop there because all of these pieces are still being multiplied together.
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I can see that I have a couple of x and a couple of y.
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Let us use our commutative property for multiplication to change the order of these parts.
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x³ x⁸ y⁶ y^16.
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The reason why I’m doing this is so we can better see that I need to apply one more of our rules that will be the product rule.
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Both of these have the same base and I can simply add their exponents together, the 3 and 8.
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-27 x^(3+8) is 11 and we can do the same thing down here with our y since they have the same base as well and are being multiplied.
00:25:30.700 --> 00:25:37.700
y^(6+16) would be 22.
00:25:37.700 --> 00:25:43.200
We finally got down to our answer and apply that to all the rules that we could.
00:25:43.200 --> 00:25:47.900
You can be very comfortable with mixing and matching and putting these rules together.
00:25:47.900 --> 00:25:55.500
Just be very careful that you do them correctly.
00:25:55.500 --> 00:26:00.700
A very important rule that we will need is the 0 exponent rule.
00:26:00.700 --> 00:26:10.100
What this rule says is that when you raise anything to the 0 power like a over here, that what you get is simply 1.
00:26:10.100 --> 00:26:12.900
This is probably one of the most curious rules that you will come across.
00:26:12.900 --> 00:26:21.400
After all, usually when you are dealing with 0, you are familiar with multiplying by 0 and getting 0 or even adding 0 and do not change anything.
00:26:21.400 --> 00:26:27.200
Why is it that when we take something to the 0 power, it becomes 1?
00:26:27.200 --> 00:26:32.700
One reason that we do this is if we want to be consistent with the rest of our rules.
00:26:32.700 --> 00:26:40.600
After all, we have been building up a lot of other rules, combining things, our repeated multiplication, we wanted to the mesh well with those other rules.
00:26:40.600 --> 00:26:49.400
Let us see how it actually does need to be defined as 1, so that it fits with everything else we are trying to do.
00:26:49.400 --> 00:26:57.300
I’m going to look at 6⁰ × 6² and we will do this twice.
00:26:57.300 --> 00:27:02.000
The first time that I go through this, I'm going to end up using my product rule.
00:27:02.000 --> 00:27:11.200
The reason why I’m doing this is that I have the 6 and the 6 are both the same base and that rule says I need to add the exponents together.
00:27:11.200 --> 00:27:28.600
This would be 6^(0+2) now the addition is not so bad 0+2 is 2 so this would end up being 36.
00:27:28.600 --> 00:27:35.900
For this rule to stay consistent with that, I need it to also equal 36.
00:27:35.900 --> 00:27:46.100
You can see that if I look at just 6² by itself, this guy over here that does equal 36.
00:27:46.100 --> 00:27:56.000
What should I call this thing to multiply by 36 so that my final result is 36.
00:27:56.000 --> 00:28:03.500
You should not have to think on it to long, there is only one thing that I can call 6⁰.
00:28:03.500 --> 00:28:11.400
I must call it a 1 so that when it is multiplied by that 36, it still is 36 as the final result.
00:28:11.400 --> 00:28:21.200
It is the only way that we can define something to the 0 power so that it mixes well with all the rest of our properties.
00:28:21.200 --> 00:28:27.300
Let us play around this rule a little bit.
00:28:27.300 --> 00:28:30.300
This new rule is the quotient rule.
00:28:30.300 --> 00:28:40.200
In this one we are dealing with two things that have the same base so a and a and they are being divided.
00:28:40.200 --> 00:28:46.000
When we divide like this, it looks like we can simply take their exponents and subtract them.
00:28:46.000 --> 00:28:54.700
It is a lot like our product rule only that one dealt with multiplication and addition, this one has division and subtraction.
00:28:54.700 --> 00:29:00.000
Some important things to know of course we must have the same base for this to work out.
00:29:00.000 --> 00:29:02.900
We are dealing with division and subtraction.
00:29:02.900 --> 00:29:07.100
This is a quick example to see why this might work.
00:29:07.100 --> 00:29:14.200
Let us look at 5⁴/ 5².
00:29:14.200 --> 00:29:28.400
Suppose I knew nothing about the quotient rule, one way that I could just end up dealing with this, is looking at it as repeated multiplication 5 × 5 × 5.
00:29:28.400 --> 00:29:34.500
Then I could look at the bottom and say okay what is 5²?
00:29:34.500 --> 00:29:44.000
That would be 5 × 5 and then I can go through canceling out my extra 5 in the top and in the bottom.
00:29:44.000 --> 00:29:51.700
This would mean 5 × 5, which is the same as 5².
00:29:51.700 --> 00:29:58.000
If we look at the result of this, you know what happened to the original versus the answer in the end,
00:29:58.000 --> 00:30:01.000
you can see that we also have to is subtract those exponents.
00:30:01.000 --> 00:30:04.100
4-2 does equal 2.
00:30:04.100 --> 00:30:10.300
This is the way that we will handle things that have the same base and we are dealing with division.
00:30:10.300 --> 00:30:16.300
We will simply subtract their exponents.
00:30:16.300 --> 00:30:24.800
Now, unfortunately this does lead to a very interesting situation and potentially it might give us some negative exponents.
00:30:24.800 --> 00:30:28.700
To see why this might happen, let us use an example with numbers so
00:30:28.700 --> 00:30:33.400
we get a good sense of some of the strange things that we want to be prepared for.
00:30:33.400 --> 00:30:38.600
This one I have 2⁴ ÷ 2⁵.
00:30:38.600 --> 00:30:45.000
The first way I’m going to handle this, is I am going to use my rule for the quotient rule.
00:30:45.000 --> 00:30:52.200
This will be 2^(4-5).
00:30:52.200 --> 00:30:58.300
If you take 4 – 5, you will get 2⁻¹.
00:30:58.300 --> 00:31:06.800
Now I can see that is what the product rule tells me to do, but you know what exactly does that mean to have 2⁻¹.
00:31:06.800 --> 00:31:15.000
After all, when we are dealing with just whole numbers, then I would often look at this as repeated multiplication, but how can I multiply 2 by itself?
00:31:15.000 --> 00:31:19.300
A -1 number of times and that just does not seem to make sense.
00:31:19.300 --> 00:31:25.900
To get a handle of what that means, I'm going to look at the original problem as repeated multiplication.
00:31:25.900 --> 00:31:37.800
This would be 2 × 2 × 2 × 2 all over 2 × 2 × 2 × 2 × 2.
00:31:37.800 --> 00:31:42.200
I have 4 of them on top and 5 of them on the bottom.
00:31:42.200 --> 00:31:51.100
Let us go through and cancel out all of our extra 2s, 4 from the top and 4 from the bottom.
00:31:51.100 --> 00:31:58.500
You can see from doing this, there is only one left and it is on the bottom so this would be ½ .
00:31:58.500 --> 00:32:07.400
Now comes the important part that if we look at each sides of our work out, we have used valid rules and we have done things correctly.
00:32:07.400 --> 00:32:13.700
What I have here are two different expressions which are the same thing.
00:32:13.700 --> 00:32:18.100
This gives us a good clue on what we should do with those negative exponents.
00:32:18.100 --> 00:32:25.100
We want to interpret our negative exponents by putting the base in the denominator of our fraction.
00:32:25.100 --> 00:32:29.000
It is how we will end up handling these things.
00:32:29.000 --> 00:32:40.200
It will take a little bit more work to get comfortable with these, let us see what else we can do.
00:32:40.200 --> 00:32:49.200
The rule that we have just developed is anytime we have a base raised to a negative exponent we will put it in the denominator of the fraction
00:32:49.200 --> 00:32:55.200
and we will change that base, a rule change that exponent to a positive number.
00:32:55.200 --> 00:33:02.800
Another way to say that is if you have an expression raised to a negative power it can be rewritten in the denominator with a positive power.
00:33:02.800 --> 00:33:06.400
It leads to something very interesting.
00:33:06.400 --> 00:33:11.600
If you have a negative in the top of your fraction and a negative in the bottom,
00:33:11.600 --> 00:33:17.400
then what you can end up doing is changing the location of where those things end up.
00:33:17.400 --> 00:33:23.400
I have a^-n and it was on the top but now it is in the bottom.
00:33:23.400 --> 00:33:30.100
I had b^-n to in the bottom, now it is in the top.
00:33:30.100 --> 00:33:35.800
A good rule of thumb that I give my students to keep track of what to do with that negative in the exponent
00:33:35.800 --> 00:33:40.300
is think of it as changing the location of where something is.
00:33:40.300 --> 00:33:50.000
If the negative exponent is in the numerator, think of the top, go ahead and move it to the bottom at your denominator and make it positive.
00:33:50.000 --> 00:34:00.000
The other situation, if it is already in the bottom at your denominator, then move it to the numerator the top and make it positive.
00:34:00.000 --> 00:34:07.100
It will help you handle these in a nice quick way.
00:34:07.100 --> 00:34:15.600
Now that we know a lot more about our exponents and especially those negative ones, let us get into using them a bit more.
00:34:15.600 --> 00:34:22.600
Here I have ¼⁻³.
00:34:22.600 --> 00:34:30.000
One of the first rules that I'm going to use is to spread out that -3 onto the top and bottom of my fraction.
00:34:30.000 --> 00:34:36.800
I have 1⁻³ / 4⁻³.
00:34:36.800 --> 00:34:40.300
Here is where I’m going to handle that negative exponent.
00:34:40.300 --> 00:34:49.700
That 1⁻³ on the top, I’m going to move it to the bottom and make its exponent positive.
00:34:49.700 --> 00:34:50.800
I’m going to do the same thing with a 4.
00:34:50.800 --> 00:34:58.200
4⁻³ now it will go to the top and its exponent will now be positive as well.
00:34:58.200 --> 00:35:15.800
From here, I just go through simplify it, 4³ = 64, 1³ = 1 so it is 64.
00:35:15.800 --> 00:35:22.800
Let us try the same thing with the next one and notice how things are changing location.
00:35:22.800 --> 00:35:30.100
2⁻³ will end up in the bottom as 2³.
00:35:30.100 --> 00:35:38.900
3⁻⁴ that one is going to go into the top as 3⁴.
00:35:38.900 --> 00:35:44.400
I will go ahead and simplify from here.
00:35:44.400 --> 00:35:54.700
There may be a few situations where some things will have positive exponents and some things will have negative exponents.
00:35:54.700 --> 00:36:00.300
The only ones that will change locations will be the ones with negative exponents.
00:36:00.300 --> 00:36:11.800
If I have an example like x² and y⁻³, then the x² will remain in the same spot but the y must go into the bottom since it had the negative exponent.
00:36:11.800 --> 00:36:17.300
Watch out for those.
00:36:17.300 --> 00:36:24.300
Let us see if we can definitely combine many more of our rules together and simplify each of these expressions.
00:36:24.300 --> 00:36:31.500
We got lots of rules to keep track of so we will just do this carefully, bit by bit.
00:36:31.500 --> 00:36:38.600
In this first one I have the (x² / 2y³)⁻³.
00:36:38.600 --> 00:36:43.700
One of the biggest features I can see here is probably that fraction.
00:36:43.700 --> 00:36:49.500
I’m going to give the -3 to the top and the bottom of my fraction.
00:36:49.500 --> 00:36:58.400
I have x² and will give it -3 and I will do the same thing with the bottom.
00:36:58.400 --> 00:37:05.200
Now one thing I can see on the top is I have x² and that is being in turn raised to another power.
00:37:05.200 --> 00:37:13.200
Our rule for that says we need to multiply the exponents x⁻⁶.
00:37:13.200 --> 00:37:14.400
What to do with the bottom?
00:37:14.400 --> 00:37:19.600
In the bottom we have some multiplication in there, so 2 × y³.
00:37:19.600 --> 00:37:29.800
I need to give that -3 to each of the pieces down here.
00:37:29.800 --> 00:37:37.800
Moving on, bit by bit, I will deal with that negative on the x and eventually that negative on the 2.
00:37:37.800 --> 00:37:42.600
Let us see what we need to do with the y³ and y⁻³.
00:37:42.600 --> 00:37:49.200
The rule says I need to multiply those things together and get -9.
00:37:49.200 --> 00:37:53.300
I think I have used all of my product rules and power rules and quotient rule.
00:37:53.300 --> 00:37:56.900
It is time to use that negative exponent rule.
00:37:56.900 --> 00:38:01.300
Anything that has a negative exponent on it is on the top and I’m going to put in the bottom.
00:38:01.300 --> 00:38:06.800
If it was on the bottom, it is going right to the top.
00:38:06.800 --> 00:38:11.700
In the bottom, x⁶ and that guy is done.
00:38:11.700 --> 00:38:18.300
In the top, 2³ and y⁹.
00:38:18.300 --> 00:38:23.600
One last thing to go ahead and clean this up, 2³ is 8.
00:38:23.600 --> 00:38:27.300
I will just ahead and put it in there.
00:38:27.300 --> 00:38:33.700
Our final simplified expression for this one is 8y⁹ / x⁶.
00:38:33.700 --> 00:38:37.900
It is quite a bit of work, but it is what happens.
00:38:37.900 --> 00:38:40.300
Let us do another one.
00:38:40.300 --> 00:38:47.500
This one is for 4h⁻⁵ / m – (2 × k).
00:38:47.500 --> 00:38:56.000
I do not see a whole lot of rules in terms of spreading things out over multiplication, but one thing I do want to do is take care of those negative exponents.
00:38:56.000 --> 00:39:01.500
I have one of them up here on the h and another one here on the m.
00:39:01.500 --> 00:39:08.100
Let me first write down the things that do not have negative exponents, they will be in exactly the same spot.
00:39:08.100 --> 00:39:14.300
The h⁻⁵ I need to put that in the bottom as h⁵.
00:39:14.300 --> 00:39:20.600
The m⁻² it now needs to go to the top as m².
00:39:20.600 --> 00:39:28.600
Okay, I would continue simplifying from here if I could but I think I do not see anything else that has the same base.
00:39:28.600 --> 00:39:35.900
I will leave it as it is and will call this one done.
00:39:35.900 --> 00:39:41.600
It is time to put all of our rules in practice and see if we can do a much harder problem.
00:39:41.600 --> 00:39:55.300
This one is (3⁹ × x² × y)⁻² / 3³ × x⁻⁴ × y.
00:39:55.300 --> 00:40:02.800
One thing I want to point out with this one is that when you are applying the rules, you do have a little bit of freedom on which ones you do first.
00:40:02.800 --> 00:40:08.600
Experiment with trying the rules in a different order and see if you come up with the same answer.
00:40:08.600 --> 00:40:14.500
As long as you apply the rules carefully and correctly, it should work out just fine.
00:40:14.500 --> 00:40:16.800
What shall we do with this one?
00:40:16.800 --> 00:40:25.600
I'm going to go ahead and take that -2 and spread it out on my x² and y since both of those are being multiplied on the inside there.
00:40:25.600 --> 00:40:33.800
3⁹ now have an (x²)⁻².
00:40:33.800 --> 00:40:37.900
I also have a y⁻² as well.
00:40:37.900 --> 00:40:48.000
I will put that -2 in both of them.
00:40:48.000 --> 00:41:10.700
Let us go ahead and multiply our exponents here, 3⁹ then 2 × -2 would be x⁻⁴ y⁻².
00:41:10.700 --> 00:41:12.400
Let us see where can I go from here.
00:41:12.400 --> 00:41:19.400
I like dealing with those negative exponents and changing the location of things so I do not have to worry about negatives.
00:41:19.400 --> 00:41:21.400
Let us do that first.
00:41:21.400 --> 00:41:26.000
Everywhere I see a negative exponent that part will change its location.
00:41:26.000 --> 00:41:33.000
I’m going to leave the 3⁹ for now and leave my 3³ and let us take care of this one.
00:41:33.000 --> 00:41:38.900
x⁻⁴ will be x⁴ in the bottom.
00:41:38.900 --> 00:41:45.200
This one which is x⁻⁴, let us put that in the top.
00:41:45.200 --> 00:41:55.700
y⁻² is in the bottom and this y already has a positive exponent, so no need to change that one.
00:41:55.700 --> 00:41:59.200
Continuing on, let us see what else we can do.
00:41:59.200 --> 00:42:09.600
These 3s out here, they have exactly the same base so I can subtract their exponents, 9 – 3.
00:42:09.600 --> 00:42:21.000
I can do the exact same thing with these x’s, subtract their exponents.
00:42:21.000 --> 00:42:23.400
What should I do with those y’s on the bottom?
00:42:23.400 --> 00:42:30.000
They have the same base, so I will add those exponents together.
00:42:30.000 --> 00:42:33.800
Good way to start crunching things down.
00:42:33.800 --> 00:42:44.300
This will be 3⁶ x⁰ / y³.
00:42:44.300 --> 00:43:00.900
3⁶ = 729 x⁰, remember that is one of our special ones is 1, all over y³.
00:43:00.900 --> 00:43:09.600
This one crunches down quite a bit, but in the end we are left with 729 / y³.
00:43:09.600 --> 00:43:17.100
We do not have anything else to combine and so we will consider this one simplified.
00:43:17.100 --> 00:43:24.300
Only one more to do this one, we want to make an expression that represents the area of the figure.
00:43:24.300 --> 00:43:29.300
I want something that is a little bit more like a word problem, at least something that we have to drive out of.
00:43:29.300 --> 00:43:33.500
We will simply use our rules along the way to crunch this down.
00:43:33.500 --> 00:43:44.900
When I look at the area of a rectangle like this, it is formed by taking the width of that rectangle and multiplying it by the length.
00:43:44.900 --> 00:43:50.600
I’m not sure what is the width and length here, since both of them are written as expressions.
00:43:50.600 --> 00:44:00.600
I will write those in, area is the width 4x² and the length is 8x⁴.
00:44:00.600 --> 00:44:06.600
That means I can take this expression for the area and put it together using some of my rules.
00:44:06.600 --> 00:44:07.700
Let us rearrange that.
00:44:07.700 --> 00:44:17.400
I’m going to put my numbers 4 and 8 together, and my variables and exponents together.
00:44:17.400 --> 00:44:24.700
4 × 8 = 32 and I have x² and x⁴.
00:44:24.700 --> 00:44:31.300
I can do these by adding them together since they have the same base giving me x⁶.
00:44:31.300 --> 00:44:37.200
This expression would be equal to the area of the rectangle.
00:44:37.200 --> 00:44:43.500
It has quite a lot of rules to enjoy but again with a little bit of extra practice, they should become a little more familiar.
00:44:43.500 --> 00:44:48.900
Watch for those special ones like raising something to the 0 power so that you know that is always 1.
00:44:48.900 --> 00:44:51.000
Thank you for watching www.educator.com.