 For notation of multiplication and division, I'm going to be using a tool called dimensional analysis. The logic of dimensional analysis is like this. I split multiplication and division across a horizontal line. Multiplication is written in the numerator, division is written in the denominator, and then I separate each multiplication or division operator with a vertical line. This. So if I wanted to represent a concept like 3 multiplied by 4 multiplied by 2 divided by 8, I would write that as 3 vertical line, 4 vertical line, 2 vertical line. Note that those are all written in the numerator because they are all multiplied in this part of the analysis. Then I'm dividing by 8 and I would write that in the denominator. This makes it convenient for me to keep track of all the steps in my calculation. It's pretty easy for this particular calculation, but as I start to have more and more complicated calculations, this can make it easier for me to keep track of what's happening. So just for completion's sake, we can calculate that by taking 3 multiplied by 4 multiplied by 2 divided by 8, and we get 3. Whoo! Where dimensional analysis gets particularly powerful is when you start introducing units into your calculation. Especially if you are trying to convert your calculation to a specific unit at the end or if you're trying to handle unit conversions along the way. So instead, let's consider one half multiplied by let's try that again, one half multiplied by 30 kilograms multiplied by 50 miles an hour. And let's square that velocity just for good measure. That quantity that I get at the end of this calculation will be a representation of energy. I know that because I'm taking one half times mass times velocity squared, which is going to result in a quantity that is mass distance squared per time squared. I might want to represent it in a different unit of energy other than kilogram mile per hour squared. For that, it might be easier for me to keep track of the conversion with dimensional analysis. So I would write that by taking one half, which I could write as 0.5 or one divided by two, then I'm multiplying by 30 kilograms, and then I'm multiplying again by 50 miles per hour. At this point, I could write another vertical line and then another 50 and another miles and another hour and that would be completely fine or because each of these is appearing twice, I could just square all of my terms in the first quantity. 50 squared mile squared per hour squared. So if I were to just take one half times 30 times 50 squared, the result would be an expression of energy in the unit kilogram mile squared per hour squared, which is a valid representation of energy, but not particularly useful. Let's say I wanted to get a result in kilojoules. For that, I would have to figure out the unit conversion associated with going from kilogram mile squared per hour squared to kilojoules. But I probably don't have that conversion memorized. That's a very specific conversion. Instead, I can handle each of the conversions individually. Those are a little bit more straightforward and I have a conversion factor sheet. Full of all the common ones. Armed with that conversion factor sheet, I should be able to handle this unit conversion in a pretty straightforward manner. As a general rule, I would encourage you to approach a unit conversion within dimensional analysis by starting at your destination and working backwards. You could completely approach it by trying to convert one step at a time until you end up at your destination, but if you get lost along the way, sometimes you end up doing additional unit conversions that aren't helpful or aren't necessary, whereas starting at your destination and working backwards, breaking all of the derived units into their primary units, you will eventually cancel everything out. So let's try converting from kilojoules down into its component parts. Since kilojoules is what I want, I'll write that in the numerator, so it will be left in the numerator after I cancel all my other units. And I will break it down into its primary components by writing a kilojoule is equal to a thousand joules. Now, why can I do that? Well, in dimensional analysis, I'm just multiplying stuff together, right? And it would be completely valid for me to write multiplied by one at the end of a calculation. A calculation multiplied by one is still the same result. And I could write one in a number of different ways. I could write it as one over one, one divided by one is still one, or I could write it as three divided by three, which is still one, or I could write it by saying three divided by two times six divided by four. This quantity is equivalent to this quantity, therefore I'm still multiplying by one. The same goes for unit conversion. As long as I write equivalent quantities in the numerator and denominator, I'm just multiplying by one. So because one kilojoule is equivalent to one thousand joules, this quantity here is just multiplying by one. Now let's keep multiplying by one. I know that a joule is one Newton times a meter. If I didn't know that, I'll stop my head. I can see in my conversion factor sheet that one joule is equal to one Newton times a meter, and then I could write one Newton is equivalent to one kilogram meter per second squared, and at this point Newtons will cancel Newtons, joules will cancel joules, and kilograms will cancel kilograms. So in order to be left with kilojoules in the numerator and nothing else, I need to get meters and meters in the denominator to cancel miles squared in the numerator. Well, I know one mile is 5,280 feet. I know that there are 3.280 feet in a meter, 3.280 feet in a meter, and if I square each of these terms, I'm still multiplying by one. So it's just fine. Feet squared cancels feet squared, miles squared cancels miles squared, and meters and meters cancel meters squared. Now, second squared and hour squared. Well, I know one hour is equivalent to 60 minutes, and I know one minute is equivalent to 60 seconds. And again, if I square everything, I'm still multiplying by one. That's not a square, that's a squiggly. One minute, about about 60 seconds, quantity squared. One squared is one. It's nice and boring. Hour squared cancels hour squared. Minute squared cancels minute squared. Second squared cancels second squared. All of my units have now canceled, leaving me with kilojoules in the numerator. So if I take one half times 30 times 50 squared times 5,280 squared divided by 1,000 times 3.2808 squared times 60 squared times 60 squared, I will get an answer in kilojoules. But for that, let's go back to the calculator. And we will type one times 30 times 50 squared. Come on calculator, you can do it. Times 5,280 squared. That's everything in the numerator line because one squared is just one. One squared is just one. Then I divide by two times 1,000 times 3.2808 squared times 60 squared times 60 squared. And I get 7.49. And again, that result would be in kilojoules. So it seems a bit intimidating when you write it out like this. But it's a much more organized way to keep track of your units and the calculation steps than trying to handle the unit conversion in a form like this.