Bam!! Mr. Tarrou! Okay. We are going to be looking at evaluating algebraic expressions. An evaluating means that when we get done we are going to have just a singular or single numerical value, generally. And we are going to use values of square root of 5 for W, -3 for X, and 2 for Y, and 4 for Z. We are going to run through six examples. For all of these examples since we are giving these numerical values to plug into these expressions, we are going to be careful of parentheses to make sure our signs end up being correct when we are done, and we are going to remember the order of operations. You might have a phrase that you use to help you remember that order of operations. My phrase that I use is "Please Excuse My Dear Aunt Sally" And the first letter is what these operations are. Please for parentheses; Excuse for exponents, My Dear for multiplication and division, and Aunt Sally for addition and subtraction. Now when you look at algebraic expression you are supposed to simplify what is inside the parenthesis first, if possible, and these are all going to be numerical value, so we will. Then deal with you exponents and then work on your multiplication and division. And you do multiplication and division, and actually as you look at this expression you read them like a book. You read them from left to right. Now I can't say two words at one time so it just comes out multiply and divide, but they are on the same sort of like level playing field. We don't really multiply before we divide or vice versa. We read these expressions left to right and whatever math function we come up with first, take care of that. So if multiplication comes up first, then multiply first; if division comes up first, then you are going to divide first Like 10 divided by 2 times , let's say, 3. Well if you were stuck in your mind, that you have to multiply before you divide you might think that you are going to do 2 x 3, which is 6 and then think that 10 divided by 6, which is 10/6 or that would be 5/3 Well, that is not correct. None of this is. You are going to look at multiplication and division and read that expression left to right and do whatever comes first. So as I read this left to right, I have 10 divided by 2, which is 5 and then I bring down the multiplication of 3 and 5 x 3 is equal to 15. So, multiplication and division, you don't multiply first, you go left to right to the expression and you do whatever math operation you see first as far as that pairing goes along. Same thing with addition and subtraction. We don't really necessarily add before we subtract. Those are on the same level playing field. You read through the expression left to right and you do whatever of these two comes first. now, of course, ultimately the parentheses first, then it is exponents, then it is multiplication and division together, whatever comes first, then it is addition and subtraction. Now, keeping that in mind. Let's do our first example. Actually it is our second now right. we have 3x^2+y-7. Okay so that's are expression; we are going to come to the list of numeric values that we are asked to plug into it. And we are going to have 3 parentheses X squared; so that is going to be 3 (-3) squared + Y, now plus (2) -7. And I am going to do these math operations one step at a time, and rewrite the whole new line because the more I see students, especially the ones at the beginning of algebra that short-cut their work, the more likely they are to make careless mistakes. They generally have an idea of what they are doing but consistently get the wrong answer. So, parentheses there is nothing to do inside here (-3)and there is nothing to do inside here (2); so then we go into exponents: -3 squared is 3 x -3 x -3. There is no reason for you to do this in your scratch work. I am just teaching you -3 squared is -3 times -3. And we get 9. And I don't need this parenthesis around 2 because it is just a plus sign out front, so I have dropped them. And we have got plus 2 minus 7. 3 times 9, keeping with the order of operations multiplication next, 3 x 9 = 27 +2 - 7, kind of a funny looking 7, So we have now, 27 + 2 is 29 minus 7, which ultimately comes out to be 22. Now, your teacher I am sure is going to be asking you to show some work. You may not have to show this much, but, you know, at least show a few lines to let the teacher know that you are actually doing the order of operations and you are not just typing these values into your two-line scientific calculator or your graphing calculator that does your order of operations for you and come down to the answer of 22. You might be noticing that not only am I doing one step at a time, but I am using parentheses when I substitute in these numerical values. and here is the reason why. There is a difference just as an example, you just saw me write (-3)squared. Well, there is a difference between (-3) squared and -3 squared. These expressions are not equal to each other. (-3)squared is (-3)(-3), as you just saw me write, and it comes out to be 9. Well, over here where it is -3 squared, exponents only act on what they are touching or sitting upon. So when it is -3 squared without the parentheses, the only thing being squared is the 3. This actually can be rewritten as -1 x 3 squared. When you have a negative sign that can be written as a multiplication of -1 and again,it is only the 3 being squared, so you have -1 x 3 squared is 3x3 and that comes out to be -9. Depending on what side of the calculator you are using, this difference may have been hidden for quite a while. So you do need to be aware that these parentheses do serve a point, and you cannot understand that for quite a while and depending on whether you are doing these math operations in your head or maybe just an older style scientific calculator with one line, this difference might have been hidden, but as soon as you start using those two line scientific or graphic calculators that really pay attention to your syntax, that is how you type these expressions and these values into your calculator, you will start noticing if you dont understand this relationship here, this difference, you will start getting some incorrect answers and not understand or realize why. All right our next example, and these will increase in difficulty as we go. You have (YZ - X)squared. Okay, so we've got Y, which is 2, and Z, which is 4. And if I write it like that 24, it shows 2 X 4, but if you are in a hurry and you miss that little dot it can still be misconstrued as a 24, if you are doing a test and you are nervous, so make sure that you use parentheses with everything that you plug in (2)(4), that is why I like them. (2)(4) - X, which is -3 . Here is another good reason (2)(4)-(-3) for using parentheses when you do substitution. My expression already has a negative sign and I am plugging in a number that has a negative sign. A lot of students will forget that, you know, that there is a negative sign already there and I am plugging in a number that is negative and they forget one of them and end up, you know, having a sign error in their problem. So, make sure again that everywhere that you are going to plug in a number, put those parentheses there, even before you plug in the numbers. Actually, this is a good idea. Okay ((Y( Z) - (X))squared. Don't even think about the numbers you are plugging; just write out the pattern and put in a set of parentheses everywhere that there is a variable and then go in and plug in the numbers- Y is 2; Z is 4; and then with the negative already there and the negative number that you are plugging in, there won't be any questions about "Did I already put that negative sign there, but is there another one coming in?" Okay, anyway, we have parentheses and we a lot of stuff in those parentheses, so we are going to simplify (2)(4) is 8. A negative times a negative or when two negatives are together, and it is multiplication, is effectively -1 x -3 so that is positive 3... (8+ 3) squared and 8+3 = 11 (11)squared is 121 . So there you go . Okay, our next example. We have Y squared over X+Z - X squared over Y - X. You can see that we are stepping up the difficulty over here. Y squared, so (2)^2 over X which is -3 + Z which is 4. Some of these parentheses that I am putting in are not really going to do anything; it is just habit X is -3 over Y, which is 2 -X, which is -3 again, so let's work this out and see what happens 2 squared is equal to 4. I am going to do a couple things at once here. Fraction bars act as a grouping symbol. It is kind of like, actually it is not kind of like, it is a grouping symbol. It does act as a parentheses in the sense that you need to get the top and the bottom simplified to a single number or simplified as much as possible before you actually try to do the division -3 + 4 is -1, excuse me, +1 minus -3 squared, it is going to be -3 times -3 because it is in parenthesis. That is 9, over negative and a negative, that is going to make a positive so that is going to be 2+3 because of these negatives coming together and making a positive value. So we have 4 over 1 minus 9 over 5. So I have some fractions left that I have to deal with Well, let's not forget. You cannot divide fractions unless you have a common denominator. So So 4 minus 9 or 4 over 1 minus 9 over 5 is not., you know. I am not going to subtract the tops and the bottoms. I am going to make common denominators first. So I am looking for the .lowest common multiple between 1 and 5. What is the smallest number that 1 and 5 can both go into, well that is going to be 5. So I am going to multiply the top and the bottom by 5 This is going to become 5 X 4, which is 20 over 5; 5x1 minus 9/5. Now that I have my common denominators I got by multiplying the top and the bottom of the fraction of the same value, which is essentially one. 5 divided by 5 is 1. I am just making this fraction look different; I am not changing the value. Once I have the common denominator, I will subtract the top number and I am done. 20 -9 is 11 over 5. Now that is done and it is an improper fraction but if the teacher has been teaching mixed fractions where they do not want the top number to be larger than the bottom number, the numerator to be larger than the denominator then you are going to need to turn 11/5 into a mixed fraction. Well you do that by division You know, how many 5s are there in 11? What is 11 divided by 5? In a calculator that is 2.2 but I don't want decimals. How many 5's are in 11? 5, 10 - so that's 2. What's the remainder? 5 x 2 = 10 The numerator was 11, so if I take out the 10 from the 11, that gives me a remainder of 1. So use whatever form of the fraction your teacher wants, improper or mixed , and you'll be good. A few more examples and we will call it a day for this video; hopefully, you are finding it helpful. Evaluating expressions, we got W squared minus 3X, so that is going to be ( ) something squared minus 3 ( ) times something well W is square root of 5, and the X is equal to -3, and we have to work this out. Well, parentheses , nothing to do in there and nothing to do in there. Exponents. Square root of 5 squared is the square root of 5 times the square root of 5. And the square root of 5 times the square root of 5, you multiply them together 5 times 5 is 25 and the square root of 25 is 5. Okay, so that is this little piece. -3 times -3 is, when you multiply two negatives, is a positive answer. When you multiply two positives you get a positive answer. But these are both negative. -3 times -3 is positive 9. and 5 plus 9 is 14. Last Example. We have got the absolute value of 3X plus Z squared. All right, absolute value If you don't know what these absolute value symbols are, if you have not covered those yet in your math class, don't worry about it. If you have seen them before, one thing that is easy to do is just drop the negative sign. Okay, the absolute value of 7 is 7 and there is no negative sign there. And the absolute about -7 is also equals 7. Well, what the absolute value symbol does by dropping the negative sign in front of a single numerical value is that it tells you how far that point is away from the origin. Seven is seven units away from the origin, seven units to the right. Negative 7 is seven units from the origin as well zero. It is seven units to the left. It just appears that you are dropping the negative sign if there is one, but the absolute value function tells you how far the number is away from zero I think it is from the origin before, but zero on the number line. But we cannot apply the absolute value function until the inside, simplified into a single number. They are, after all, they look like parentheses and they act like parentheses; they are a grouping symbol. It is just that at the end, if the number happens to be negative you are going to drop it and say "This is my final answer". So let's see what happens. We have the absolute value of 3 times what X is, plus whatever Z is squared and closed with the absolute value sign. X is negative 3 again; Z is 4; okay so we have 3 times -3 is -9, and I want to do two operations at once. This is multiplication and I am also going to I said you are supposed to do parenthesis and then exponents. So you really are supposed to do exponents before you multiply, but I am going to do both of those steps in the same line here. 4 squared, 4 squared, 4 times 4, not 4 times 2. 4 times 4 is 16. And as a cautionary note here, If I try to apply the absolute value symbol for function right now and make the 9 positive,I am going to get a wrong answer. You do not apply the absolute value function until the inside is cleaned up or simplified or evaluated into a single value, so we've got -9 plus 16 when your numbers have different signs, you are going to subtract those numbers and keep the sign of the largest number. The absolute value of 16 is bigger than the absolute value of 9, so 16 minus 9 is 7 and the absolute value of 7 , oh, I didn't do that on purpose., ends up being 7 and there is our answer. Well, I am Mr. Tarrou. BAM! Go Do Your Homework:) Thank you for watching. I hope this has helped.