chapter p1 we're going to be going over algebraic expressions mathematical models and the set of real numbers so before we get into exponential notation just the definition of a variable here up top a variable is just a letter used to represent various numbers and typically we use X or Y it doesn't have to be X or Y we can use just about anything so getting into exponential notation exponential notation is going to be what we have here B to the N so this B is the base which is why I'm writing it as a B the N up here is the exponent so it's a superscript it's the exponent or the power that it's the base is being raised to so what that means is B to the N is going to be B the base being multiplied n times the exponent number of times and like I just read it B to the nth power or B to the N and an important note here which we'll go over in a little bit B to the 0 power equals 1 and B to the first power just equals B ok so some examples of this let's say we had start with 7 to the 0 so 7 to the 0 what I just mentioned B to the 0 anything to the 0 other than 0 is equal to 1 let's do 9 to the first power that's equal to 9 and we'll do 8 to the second power so based on the definition up here that means 8 times 8 8 times 8 is 64 so 8 to the 2nd power we multiply it together twice and when we're when we have something to the second power we can say that that is squared 8 squared let's do five to the third power so that would be five being multiplied five times so 5 times 5 times 5 is 125 when we raise something to the third power we can say cubed so that would be 5 cubed and maybe one more here 2 to the fourth 2 being multiplied 4 times and we get 16 so squared and cube there just our kind of shorthand ones because they tend to pop up the most for us so the next thing for an algebraic expression an algebraic expression is going to be just an expression so no equal sign which can consist of variables numbers addition subtraction multiplication division and maybe some exponents or roots so for example something like 3x plus 6 this is an expression there's no equal sign or something like x squared minus 3 that's also an expression so we're gonna be evaluating expressions in order to evaluate expressions we need the correct order of operations so we need to evaluate them in this order so we're gonna do whatever is inside the parentheses first if there is any then we're gonna do the exponents and then we're either gonna do multiplication or division whichever one comes first from left to right and then same thing after that we're gonna do addition or subtraction whichever one comes first from left to right so if we look at the first letter of each word here that is PEMDAS so you can use that if to help you remember the order of operations and just remember that the M and D multiplication division are essentially the same whichever one comes first left to right same thing with addition and subtraction let's do a couple examples here so use the order of operations to simplify each one so 8 squared minus 16 divided by 2 squared times 4 minus 3 so we would have to do parentheses first but there aren't any so exponents are first for this one 8 squared is 8 times 8 which is 64 and then we have 2 squared so I'm going to ignore everything else 2 squared is 4 ok so next would be multiplication or division whichever comes first from left to right so we have a division and multiplication but we're gonna do 16 divided by 4 first 16 divided by 4 is 4 so now we're going to do the multiplication 4 times 4 is 16 [Music] and now we just have subtraction so it's going to be 64 minus 16 so I'm going to put this up here so 64 minus 16 is 4 D 8 minus that 348 minus 3 is 45 so our answer would be 45 next for B we have 8 minus 3 and then we see brackets and parentheses so brackets are just the same thing as parentheses we just tend to use them if we have more than one set of parentheses so we have 8 minus 3 and then times negative 2 times 2 minus 5 minus 4 times 8 minus 6 so parentheses first so we go to the bracket we want to evaluate what's inside the bracket first there's parentheses inside there so that means we have to evaluate this set first and this set first so it's gonna be 8 minus 3 times negative 2 2 minus 5 is going to be negative 3 minus 4 times 8 minus 6 is 2 so now we're going to evaluate what's inside the bracket so we have multiplication in there and subtraction so we're going to do the multiplication first so negative 2 times negative 3 is positive 6 minus 4 times 2 is 8 okay and then we're going to do that bracket which is just parenthesis 6 minus 8 is going to be negative 2 multiplication next negative 3 times negative 2 is going to be plus 6 so this expression is going to equal 14 and 4 another one here we have 10 divided by 2 plus 3 times 4 divided by 12 minus 3 times 2 squared so I'm going to deal with the numerator and denominator separately so the numerator we have division and multiplication in addition so division first because it pops up first left to right 10 divided by 2 is 5 so 5 plus 3 times 4 in this denominator we have multi have a we have an exponent but we have parenthesis so we need to evaluate that first inside the parenthesis we have multiplication so we have to evaluate 3 times 2 which is 6 ok so now for the numerator again 3 times 4 is 12 denominator we have to do the parenthesis still finish them up 12 minus 6 is 6 so it's a good habit to leave parentheses around something if it's raised to an exponent even when you're done it won't matter with positive numbers but it will with negative numbers so it's good practice to leave it there so 5 + 12 17 6 squared is 6 times 6 so 36 so I'm just going to leave that just as 17 over 36 okay so let's make these a little bit harder so we're gonna be evaluating algebraic expressions now so that means we're gonna be finding the value of an expression for a given value of the variable so all we have to do is plug it in plug in whatever number and then use the proper order of operations again so let's do his first one so let's evaluate 7 plus 5 times X minus 4 cubed when X is negative when X is 6 x equals 6 so I'm gonna plug that in I'm gonna plug in a 6 into X so 6 minus 4 cubed and now it's the same exact thing that we were just doing so order of operations parentheses first 6 minus 4 is 2 so next is going to be the exponent 2 cubed is 2 times 2 times 2 which is 8 7 plus 5 times 8 we need to do the multiplication 5 times 8 is 40 and 7 plus 40 is 47 and next we have 6 times 23 minus x squared minus 2 and we're going to evaluate that when X is 13 so let's plug in a a 13 into X so 23 minus 13 squared minus 2 parentheses first 23 minus 13 is 10 exponent next 10 squared is 10 times 10 so 100 multiplication 6 times 10 600 and 600 minus 2 is 598 okay all right making these even a little bit more challenging there's something that we do have to watch out for when we have a so this note up here a negative number if it's raised to an odd power you're going to get a negative number as the result if the negative is raised to an even power you're gonna get a positive number so let's see what I mean by that so let's evaluate x squared for x equals negative 2 so when we're plugging this in we need parentheses around it negative 2 squared so negative 2 squared we know that is negative 2 times negative 2 right just based off of exponents negative times a negative is positive so 2 times 2 is 4 so it equals positive 4 so that's a negative to an even power it became positive answer so it happens for the next one X cubed let's evaluate X cubed when X is negative 3 so that's going to be negative 3 cubed we know that means negative 3 multiplied together 3 times so it was raised to the power of 3 so what that means is that we're gonna have a leftover negative when we multiply so our answer is going to be negative so the first two negatives will make it positive but the odd one out will make it negative and 3 times 3 times 3 is 27 so negative 3 cubed is negative 27 so a negative to an odd power is negative answer next one X to the fifth for x equals negative 1 so it's going to be negative 1 to the fifth so we could write this out as negative 1 times itself 5 times or we can recognize that 1 or that the exponent is odd and the base is negative so we know it's gonna be negative answer 1 raised up to the 5th power is just 1 so negative 1 raised up to an odd power as negative 1 negative 1 raised up to an even power would be positive 1 and for X to the 4th evaluate X to the 4th for x equals negative 2 so that's gonna be negative 2 to the 4th so right away I see that it's raised to an even power so we know it's gonna be positive 2 to the 4th is 2 times 2 times 2 times 2 which is 16 so if you have to write this out in expanded form being multiplied that's fine I'm just skipping it for now and for the next one we have X plus 4 squared minus 3 for x equals negative 5 so we're gonna plug that in negative 5 plus 4 squared minus 3 simplify parentheses first that's negative 1 squared minus 3 negative 1 squared that's a negative 2 an even exponent 1 times 1 is just 1 so that's 1 minus 3 which is negative 2 and for the last one here 2 times negative X minus 3 cubed 4x equals negative 1 so this is going to be plugging in a 1 into X so negative 1 minus 3 cubed inside here negative 1 minus 3 is negative 4 cubed negative 4 it's a negative 2 an odd meaning it's going to be a negative answer negative 4 times negative 4 times negative 4 is gonna be negative 64 negative 64 times 2 is gonna be negative 128 circle these getting a little crowded okay and then just an important note that I'm gonna write down here negative x squared this equals negative and then x squared meaning it's going to be negative this is not the same thing as negative x where a negative x is in parentheses and then squared because this is equal to positive x squared so parentheses 100% matter okay for the next thing so that was evaluating expressions the next thing is going to be working with equations so in equation an equation is gonna be formed simply when there's just an equal sign between two algebraic expressions so for example if we have something like 2x minus 3 equals 4x cubed so this whole entire thing is going to be an equation both sides of the equal sign are their own individual expressions okay and then the next thing here is going to be a formula definition of a formula so a formula is an equation a formula is an equation that uses variables to represent a relationship between two or more quantities so for an example of this is if we had something like f equals 9/5 times c plus 32 so what this is usually formulas are solved for one of the variables so it's solve for F here this is actually the conversion the conversion between degrees Fahrenheit and Celsius so you could plug in degrees Celsius here to solve for Fahrenheit we could solve this for Celsius if we wanted to if we solve that for C we would get five ninths times F minus 32 which will actually be getting into solving stuff like that later on hey in the process of actually coming up with formulas to describe real-world events is would be an example of mathematical modeling which is used all the time in Applied Mathematics in different disciplines let's do a couple examples here so let's use the formula this formula appear that I just wrote down to convert the given Celsius temperatures to its equivalent temperature on the Fahrenheit scale so we have essentially we're asking for the first one what is 20 degrees Celsius and Fahrenheit so what we're going to do here is we are going to plug it in so it's going to be F equals 9/5 C plus 32 so let's plug in C equals 20 so nine fifths times 20 plus 32 so all we're doing here is working with multiplying fractions first nine over five times 20 is 9 over 5 times 20 over one what we can do is we can simplify we can either multiply 9 times 20 and then divide it by 5 let's make the numbers easier let's divide 20 divided by 5 first 20 divided by 5 is going to give us 4 now it's easier to work with we can do all this whole thing in her head 9 times 4 is 36 so 36 plus 32 is 68 so this means that 20 degrees Celsius equals 68 degrees Fahrenheit let's do negative 40 degrees Celsius let's see what that is in Fahrenheit so 9/5 Celsius plus 32 we're gonna plug in negative 14 into Celsius so once again here let's let's simplify so it's a little bit easier for us so negative 40 divided by 5 is negative 8 plus 32 and 9 times 8 is going to give us negative 72 Plus that 32 so negative 72 plus 32 this is negative 40 so you're probably saying away that's the same thing that we just started with negative 40 degrees Celsius that is because negative 40 degrees Celsius is actually equal to negative 40 degrees Fahrenheit which is pretty cool if you didn't know that pretty cool how they are equal at that point ok let's do one more here so we have the formula T equals 4x squared plus 330 X plus 3310 models the average cost of tuition which is the T for public u.s. colleges for the school year ending X years after 2000 so we want to use the formula to find the average cost of tuition for the school year ending in 2012 or 2010 rather okay so we need to figure out we're looking for T so we're looking to find T when X is what when x equals what so we're trying to plug in for X so the school year ending in 2010 X was we were told represents X years after 2000 2010 is 10 years after 2000 so X here is going to be 10 so we are using this formula and we are going to plug in 10 so it's going to be many races t so it's gonna be 4 times 10 squared plus 330 times 10 plus 3310 and now just order of operations once again here so it's gonna be 10 squared is 100 I'm just gonna leave the other stuff the same for now which we could have evaluated it so I'm gonna do both of those multiplications 4 times 4 times 100 is 400 plus 300 30 times 10 is going to be 3,300 and lastly we multiply all these or we add them all together and we get 7,000 10 so this means the tuition for the school year ending in 2010 based on this mathematical model is going to be seven thousand ten dollars okay so mathematical models are typically used are typically created given previous data okay so kind of switching gears here just for a little bit we're gonna get in two sets of numbers so we're gonna be using this throughout the course a set is just a collection of objects whose contents can be clearly determined in the objects in a set or called the elements so things in a side of a set are just called elements of the set and we denote a set using braces which I call them squiggly brackets because there's squiggly brackets but they're technically called braces so two ways to represent a set here we're gonna have a the roster method and the set builder notation will pretty much be using both of them so a roster method completely just lists the alum it's out in the set and uses commas to separate them so roster method and just list them out so here we have the set of natural numbers if you remember from the previous section set of natural numbers we have just one two three four and so on these are called the counting numbers make a little note set of natural numbers and we also call these the counting numbers and I don't know if we mentioned this before but I'm gonna mention it now the dot dot dot here obviously means continues in this fashion or and so on it's called the ellipsis so it just means continues in this fashion so that's the roster method it's pretty pretty simple it's just lists everything out the next one is not as simple it's the set builder notation the elements are not listed and set builder notation but they are described so for this it looks something like what I have written right here so what we're gonna do is we read this as this first part here we start by reading it as the set of all X values this vertical line is read as such that and then everything after that is the description it tells you what you're referring to or what is in this set so this is the set of all X value such that X is a counting number less than 6 so if we were to write this in the roster method a counting number less than 6 is a natural number less than 6 which means it's going to be 1 2 3 4 and 5 so we can see how its described here but not listed out but we just listed it out using the roster method so some time at a time set builder notation will come in handy when there's too many things that we can actually physically list out okay so next thing that we're going to be doing is the intersection of sets so for the intersection of two sets we write that like this with that little cap and we read that as a intersects B it's the set of elements common to both a and B so if we were doing a Venn diagram and this was set a over here the circle was set a the circle was set B the intersection of the sets you're going to think of the word intersection normally it's where they intersect it's where they overlap so it's the sets where they it's the portion where they have in common in the middle so let's do these examples so let's find the intersection of the sets so these are all in roster method so we have the first set seven eight nine ten eleven we want to see when does that intersect six eight ten twelve so all we have to do is say well what numbers do they have in common and the numbers that they have in common are going to be they both have an eight and they both have a 10 so it's pretty simple and for the next one we have when does the set three four five six seven intersect three seven eight nine so once again what do they have in common they have a three and they both have a 7 so 3 into 7 and just make sure that you when you're writing your answer for this the answer is also a set which is why it has the squiggly brackets ok so if a set has no elements not every set has elements in it it's called the empty set or sometimes called the null set if you want to it's represented with the symbol looks like a little circle with a line diagonally through it technically there's another way to denote it it's just by drawing the squiggly brackets but not putting anything in it obviously or that one doesn't look as good to me personally so I just like using the other symbol so for example if we try to determine the intersection of these sets 2 4 6 3 5 and 7 we're saying what do they have in common they have nothing in common so we would say that this is the app the answer is the empty set the intersection of these two sets is the empty set because they have no elements in common okay so that's intersection this the next thing is the union of sets so the union of sets a and B two sets a and B we use the little cup notation looks like a U for union is going to be the set of elements that are in a or in B so it's what is in all in both of the sets essentially so essentially it's just every single thing in the sets so once again let's look at a Venn diagram here and circle a is on the left circle B is on the right so the union would be everything in both of the sets here so both of them are just shaded in so Union let's do this example let's find the union for the following seven eight nine ten and eleven for the first set we want to find the union of that with six eight ten and twelve so what we're gonna do is we are just going to list out everything that's there well one thing to keep in mind though is we don't have to repeat something if it's there in both of them so we have a six in the one on the right seven on the left they both have an 8 9 on the left they both have a 10 11 on the left and 12 on the right so Union pretty much means join right so essentially we're just joining the sets together to make one big set do two more so we have three two set three four five six seven Union three seven eight nine so I'm just gonna write everything down join the sets together for the Union they both have a three there's a four on the left five on the left six on the left they both have a seven and then eight nine on the right and for the last one here we have 1 2 5 8 13 Union 3 4 8 9 so just copying everything down again we have 1 2 3 4 5 8 9 thirteen so intersection of the sets are what they have in common the Union is joining the sets together it's everything in both sets so that brings us into some definition of other sets here so we've already done the set of natural numbers so the set of natural numbers we can denote that with a capital n oops with a capital n except with a like double backbone and that's going to be the set of numbers 1 2 3 4 5 6 7 8 and so on next one set of whole numbers which we can denote with aw with a little double backbone again so the set of whole numbers whole numbers are the natural numbers but including 0 so it's going to be 0 1 2 3 4 or 5 and so on so it's the natural numbers plus a set of zeros plus 0 so what we could say here is that the natural numbers are a subset to the whole numbers because the natural numbers are contained inside of the whole numbers next one for integers it's not denoted with an I it is denoted with a Z integers are positive or negative whole numbers so what I'm going to do for this one I'm gonna put an ellipsis on the left and say negative 4 negative 3 negative 2 negative 1 0 1 2 3 4 and so on ok so it's positive and negative whole numbers for the integers so the whole numbers are a subset of the integers because they're contained in there also next for the rational numbers so rational numbers gets a Q because the R is already taken which we'll get to in a second so rational numbers are going to be this one can't be listed out very nicely in roster method so I'm going to write this one in set builder notation so it's going to be the set of all fractions such that the numerator a and B are integers so the word ratio is in the word rational right so rational numbers are ratios of integers so for an example of this we could have something like negative 5 we can write this as negative 5 over 1 right negative 5 and 1 are both integers so whole numbers and integers so just integers integers are rational numbers because we can just place them over one another one we can have is 2/3 2 & 3 are both are both integers so this works if we think about this as a decimal this is 0.6 repeating so repeating decimals are actually rational numbers because we can rewrite them as a ratio of integers which may not seem like it but we can and if we have something just like two over five that would be 0.42 and five are obviously integers so terminating decimals like point four those are also rational numbers hey next one irrational number is going to denote this with a kind of a cursive looking I just because I don't know how to do a backbone non cursive I so for these ones I'm just gonna kind of write out the definition in words here it's going to be any number whose decimal representation is neither terminating nor repeating so irrational numbers are going to have if the you wrote them as a decimal they would go on forever so they would never stop they would keep on going but they would also never repeat so a couple commonly seen examples kind of use examples for this radical two so the square root of two which if you plug this in the calculator would be something like a squiggly equal sign here would be about one point four one four two one three five and it's gonna keep on going and it's gonna keep on going forever it's never gonna repeat itself okay another one is pi so hopefully remember pi pi is approximately three point one four one five and so on so that's also going to go on forever and it's never gonna repeat itself so radical three is irrational radical five z rational reticles 7 radical 11 so a lot of radical radicals are irrational numbers because we cannot write them as a ratio of two integers so lastly here the set of real numbers the set of real numbers so this one gets the capital R so the set of real numbers is going to be I'm going to write this in set builder notation it's going to be the set of all X's such that X is rational or irrational so rational and irrational numbers together the union of them make up the set of real numbers so almost everything that we work with is a real number any number you can think of almost as a real number the ones that are not that we'll be getting to our imaginary numbers they're not real numbers we'll worry about that at a later date and the infinities are also not real numbers okay so let's do an example to make sure we know that we're comfortable with this so let's say we have this the set of numbers here you have negative seven negative three over four zero point six repeating radical five pi seven point three and radical 81 so we're going to go through and see which one of those numbers are in all of these sets and they can obviously be in multiple sets because a lot of these sets are subsets of the other ones okay so let's do natural numbers so let's skim this for natural numbers we know natural numbers are 1 2 3 4 and so on so skimming this for one of those types of numbers we get to radical 81 radical 81 the square root of 81 is actually equal to 9 which is a natural number so that means radical 81 is a natural number we'll be getting into working with square roots more also ok so for whole numbers we know that whole numbers are natural numbers and also zero zero is in this set so this is going to be 0 and the square root of 81 for integers integers are going to be whole numbers but also including negative whole numbers there's a negative 7 up here so that's going to be negative 7 0 is an integer so is radical 81 and for rational numbers rational numbers are going to be numbers that can be written as a fraction as a ratio with the numerator and denominator being integers so this is going to be negative 7 negative 3 over 4 works zero works point 6 repeating works because we can write that as 2 over 3 radical 5 doesn't pie doesn't so 7.3 up here can be written as 73 over 10 which is a so it's a terminating decimal so we know it's a rational number can be written as a ratio of integers so that works and so does radical 81 lastly for irrational numbers that's just going to be radical five and PI okay and then for the real numbers we're going to have everything so rational numbers and irrational so I'm just going to list them all out negative seven negative three over four zero point six repeating radical five pi seven point three and radical 81 all right for the next thing so a couple more things left for the section the real number line so the real number line is essentially a graph and we use it to represent just the set of real numbers so it just looks like this I'm sure we're familiar with it we have zero here in the middle so everything to the right are positive numbers and everything to the left are negative numbers and the numbers increase from left to right so we can actually use inequalities to help us compare the values so the inequalities we have are less than less than or equal to greater than and greater than or equal to so you can see right here I have them listed out so the inequality sign opens up to the larger number and if it's or equal to we have one of the equal two slashes below it so for the first one a is less than B and then second one a less than or equal to B and then a greater than B and a greater than or equal to B so some examples of this we could have negative for less than two which we obviously know is true positive is greater than negative number could have negative three greater than or equal to negative seven so the negatives you just have to make sure you're a little bit careful negative three is greater than negative seven it's further to the right on the number line the or equal to that's completely fine so it's greater than Oh right is equal to its greater than so it suffices negative five less than or equal to zero that also is true and nine greater than or equal to nine so that last one looks a little weird but it says or equal to so technically that's true because they are equal and for the next thing here we have the absolute value which will be thinking in terms of the number line also so the absolute value of a number which we'll call X it's denoted by the vertical lines to the left and right of it so that's the absolute value of X what it is it's the distance from zero to X on the number line which is always going to either be zero or positive so technically the technical definition I wrote right here this is called a piecewise function it's chopped up in terms of the domain which we'll get to that also so it says the absolute value of x is going to equal negative x if X is negative that means if we're negating a negative it becomes positive or it equals x if X is greater than or equal to zero so it just equals the positive number if it's already positive so it looks confusing that's a technical definition for it and if we're if this isn't an expression we're simplifying we would kind of treat them like parentheses at first we would evaluate the inside of it and inside of absolute value so let's do some examples start off simple so let's rewrite each expression without the absolute value bars so we have the absolute value of 3 so this is saying how far away is 3 from 0 on the number line which is just 3 4 negative 2 this is absolute value of negative 2 how far is negative 2 away from 0 on the number line that is 2 units so all you have to do is just drop the absolute value bars and make what's inside of it positive and that's it but you should think of it as a distance from 0 for negative absolute value of 5 the absolute value of five is five but there's a negative out front order of operations says we have to keep a negative afterwards what this another way of thinking about this is negative one times the absolute value of five so this would be negative one times five which would be negative five and then for negative absolute value of negative seven the absolute value of negative seven is seven but we still have the negative out front so that's negative seven seventeen minus twenty three is inside the absolute value for Part A here so we have to simplify that first seventeen minus twenty three is negative six absolute value of negative six positive six so it's going let's make it a little bit harder here so we have the absolute value of two minus pi so we almost never unless we're doing like a word problem type thing we don't really want to use an abbreviation or an approximation for pi or irrational numbers we know that two minus pi is negative two minus pi is negative because that's about two minus 3.1 four-ish so that's going to be negative since that's negative all we have to do to come up with an equivalent expression an equivalent expression for that and drop the absolute value bars is reverse the order reverse the order and drop absolute value so what that means is this equals PI minus two so the distance from zeros are going to be the same here so for the next one the absolute value of radical three minus one so radical three minus one is actually positive it's one point something so that means we do not need to reverse the order so do not so we don't need to reverse the order but we can just drop the absolute value bars and we'll be good so this just equals radical 3-1 so there's nothing wrong with that I know it looks kind of probably looks messy but that's completely fine there's no other way of writing it where it's 100% accurate because we would have to round if we use the decimal okay one more for this particular one so we have negative 2 times 4x plus 10 plus 2 and we're going to evaluate it 4x equals negative 5 so let's plug in a negative 5 and X so it's gonna be 4 times negative 5 plus 10 plus 2 so PEMDAS order of operations we need to simplify what's inside the absolute value first there is multiplication 4 times negative 5 is negative 20 and we still need to simplify what's inside of there negative 20 plus 10 is going to be negative 10 now we have the absolute value of negative 10 which is positive 10 so it dropped the absolute value bars negative 2 times 10 negative 20 plus 2 is going to give us negative 18 pay and 4 next thing here let's simplify this following expression and this will be the last thing for this whole entire page so we have this long nasty expression with division what I'm going to do here is can simplify the numerator denominator separately so we have absolute value bars in the numerator so I'm going to simplify what's inside of those first so 12 divided by 3 times 5 times the absolute value of 2 squared plus 3 squared exponents first in side here so that's going to be 2 squared is 4 3 squared is 9 denominator we have an exponent 6 squared is 36 so it's going to be 7 plus 3 minus 36 numerator we still need to simplify what's inside those absolute values so that's going to be 4 plus 9 is 13 denominator 7 plus 3 minus 36 7 plus 3 is 10 10 minus 36 and for the numerator now we just have multiplication and division so it's going to be 12 I'll do 12 divided by 3 because it's the first one from left to right which is 4 times 5 the absolute value of 13 is just 13 10-36 for that denominator is negative 26 I'm going to bring this up here a little bit and now we have 4 times 5 times 13 so we'll see in the next section with the order of with the order of multiplying with the associative property of multiplication we can multiply these in any order so if we just did 4 times 5 we would get 20 if we multiply that by 13 next we would get 260 so 260 over negative 26 is going to give us negative 10 so our answer for this whole thing actually simplifies to just negative 10 ok so that is it for p1