units derived units are units that are made from base units and the problem with derived units is those prefixes that we know love and work so well can break down when we combine units to form things in multiple Dimensions really easy example I have my beautiful Square here my beautiful square is one cm by 1 cm by 1 cm so it's volume is 1 cm Time 1 cm Time 1 cm or 1 cubic cm I know because I've learned my prefixes that a centimeter is 10 to the2 meters so the same square that I have here is 10 the2 met by 10 thec meters by 10 The ne2 Meters so its volume is 102 M time 102 M time 10 the2 m or 10 to the6 cubic meters but wait a second the volume of the square is 1 cubic centimeter and according to the calculations we just did then 1 cubic cm must be 10 -6 cub Cub met what happened to our prefixes when you add a different dimension it changes the relationship 1 cubic cimer is indeed equal to 10 -6 cubic meters everybody writes down Phineas and Ferb Phineas and Ferb into the second dimension great movie um even though there's a scene in there that's kind of Blasphemous but um I totally digress fortunately for us the dimension gives us a clue to the new relationship when you're dealing with cubic units or square units you simply raise the linear relationship to the same power with which you're working for example if I'm dealing with cubic units like I am here 10 the 3 I take my linear relationship and I Cube it algebra moment of the day 10 to the X raised to the Y power is equal to the 10 x * y power hence -2 * 3 is -6 and it works for square units as well let's take a look at some more examples here if I have 1.39 * 10 7th cubic millimeters and I want to know how many cubic meters I have the first thing I'm going to do is I'm going to determine the relationship between cubic millimeters and cubic meters so I write the linear relationship write my linear relationship and then because I'm dealing with cubic [Music] units I'm going to cube that linear relationship so a cubic millimeter is equal to 10 to the3 * 3 -9 cubic meters once I have that relationship established I can work my unit conversion like normal 1.39 * 10 7th cubic millimeters place my cubic millimeters where they cancel out I want my cubic meters I know that 10 9 cubic met is 1 cubic millimeter multiply everything on top divide through by everything on the bottom chemistry is easy life is hard let's look at another example here I have 1 . 2 * 10 to the -5 cubic megamet and I want to know how many cubic decimeters I have and how many millimeters I have let's take them one at a time I know that a megameter is 10 6 meters so to get their cubic relationship I'm going to cube them a cubic megameter is 10 to the 6 * 3 is 18 M what about cubic deers I know a decimeter is 101 meters to get the cubic relationship I Cubit and a cubic decimeter is 10 -3rd cub meters so now I have my relationships I can go ahead and work my problem here so I have 1.2 * 105 cubic megameters I don't want cubic megameters I know the relationship between cubic megameters and meters I know that I don't want meters so I'm going to place them where they cancel out I want cubic decimeters I know that a cubic decimeter is the same thing as 103r cubic met multiply everything on top divide by everything on the bottom and I wind up with um5 um posi3 positive 16 okay how do we get to millimeters the great thing about the metric system compared to the SI system is the metric system defines a leader as a unit of volume it doesn't create a leader from something else that means that for leaders the prefixes still work so a milliliter is still 10 to the3 lers a um gigaliter is still a billion liters and that's because the metric system defines a leader as a leader whereas the SI system builds units of volume cubic meters cubic decimeters Etc from base units so those prefix break down because you're cubing things well therefore we need a conversion process to go from asymetric we need to know what the bridge is between cubic meters and liters turns out that you can be a real hit at parties instead of bringing two lers of coke you can bring two cubic decimeters you need to George Carlins a dirty word this and memorize that a cubic decimeter is the same thing as a leader you should also memorize something I told you way back at the beginning of class that a cubic centimeter is the same thing as a milliliter but you can be a real hit at parties if you begin bringing two decimeters of coke instead of just uh two liters of coke it'll get the conversation flowing trust me hey everybody write down Mickey Mouse everybody write down Mickey Mouse I didn't have a word prepared there in case you couldn't tell so if we go back to this problem that we were working on how do we get from cubic decimeters here [Music] to our millimeters I need to clean things up just a little bit here well we know that 1.2 * 10 16 cubic decimeters is the exact same thing as a liter so I ALS o have 2 1.2 * 10 the 16th lers and then life becomes easy because I know that relationship between liters and milliliters I know that 103r lers is equivalent to a milliliter so I have 1.2 * 10 19th milliliters chemistry's easy life is hard one other quick note here um take a quick look at those two examples right there pause me and try them on your own okay hopefully you paused me and tried them on your own write your linear relationship nanometer is 10 to the negative 9th meters you get the cubic relationship you Cube it -9 * 3 is -27 kilometer a kilometer is 10 to the third meters so to get to the linear relationship we to get to the cubic relationship we Cubit so a cubic kilometer is 10 to the 9th cubic meters let's keep going try these next to try these next two pause me and try them okay hopefully you've paused me and tried them let's take a look we have two different cubic metric units so we need to get at those relationships down path first we know A millimeter is 10 the -3 meters so we Cube it and we discover that a cubic millimeter is the same thing as 10 the 9th cubic meters a femtometer is 10 -5 MERS we Cube it and we get 10 the -40 F cubic meters is equal to a femtometer now we start our line 3.92 * 10 -13 cubic millimeters place our cubic millimeters where they cancel out the only thing we know is about meters 10 the 99th cubic meters place our cubic meters where they cancel out we want cubic femtometers a femtometer is 10 the 45 cubic meters multiply everything on top divide through by everything on the bottom um -22 um positive 45 [Music] is 9.2 * 10 23rd cubic feters this one may or may not have thrown you for a curve ball because you're dealing with cubic inches and cubic feet but your operations are going to remain the same you know that an inch you know that there are 12 Ines in one foot so to get that cubic relationship you Cube it the only unusual thing here is you are cubing the number 12 right so 12 12 cubed would be 144 * 12 so you wind up with 1,728 cubic inches in a cubic foot so once you have that relationship established now it's just a matter of your straightforward unit conversions so that gives you 4.7 times oh whoops just 4.7 cubic feet one last one here for you to try please 1.4 * 10 5th cm squ equals how many square kilom try this on your own hopefully you paused me again we're dealing with square units so this time when we write those relationships instead of cubing them or squaring them so a square cm is 10 -4 square meters when I do my kilometers kilometer is 10 the 3 meters squaring it so a square kilometer is 10 to the six square met so that's the big catch there um the rest is going to be straightforward how you'd work it if I didn't show you um 1.4 * 10 to the 5th everybody write down Frozen Do You Want to Build a Snowman I like that song better than I do let it go I am so over Let It Go I've let it go let it go I've let go of Let It Go Place met squared down here I want kilometers squared multiply everything on top divide through by everything on the bottom that's a positive1 - 6 is A5 chemistry is easy life is hard