# GED Math ## Метаданные - **Канал:** The Organic Chemistry Tutor - **YouTube:** https://www.youtube.com/watch?v=PyYEHq4Dc5I - **Источник:** https://ekstraktznaniy.ru/video/41225 ## Транскрипт ### Segment 1 (00:00 - 05:00) [] This video is for those of you who want to improve your score in the math section of the GED exam. Now, for each of these problems, I recommend that you pause the video and try to work on a problem yourself and then once you've done so, play the video and see if you have the right solution. So, let's start with this one. Number one, what is the perimeter of the triangle shown below? In order to calculate the perimeter of any figure, you need to add up all the sides of the figure. In this case, we have a triangle. So, we need to add up the three sides of the triangle. However, we are missing one side of the triangle. So, we need to find it. Now, you need to be familiar with something called the applies to right triangles. A right triangle is a triangle with an angle of 90° represented by the box that you see there. And here's the balite to the box. A in this case is six and we're looking for the missing side which we'll call B. So let's replace a with 6 and let's replace c with 10. 6^2 is 6 * 6 which is 36. 10^ 2 is 10 * 10 which is 100. Now let's subtract both sides by 36. 100 - 36 is 64. Now to calculate b, we need to take the square root of both sides. 64 is 8. And so that's going to be the missing side left. So let's write B is equal to 8. Now that we have the miss side left, we can calculate the perimeter, which is the sum of the three sides of the triangle. So it's A + B + C. A is equal to 6, B is 8, C is 10. Now 6 + 8 is 14. and 14 + 10 is 24. And so that is the perimeter of the triangle shown below, which means D is the right answer choice. Number two, multiply. So here we're multiplying a binomial by another binomial. A binomial is an algebraic expression that contains two terms. A monomial contains one term. A tromial contains three terms. A polomial contains many terms. So we need to use the foil method here. So let's begin by multiplying the first terms. 2x by 3x. 2 * 3 is 6. x * x is x^ 2. You need to add the exponents. 1 + 1 is 2. Next we have 2x * 2. And so that's going to be 4x -5 * 3x that's -5x5 * 2 is -10. Now our next step is to combine like terms and so those two terms are similar. You can't combine these two terms because x and x^2 are different. 4x - 15x is -1x. And so the answer is 6 x^2 - 11x - 10. And so that correlates to answer choice B. Number three, which of the following is equivalent to the expression shown below. So we have 2 over 5 + 4 7. How can we add these two fractions? What we need to do is get common denominators. The top number is the numerator and the bottom number is the denominator. And right now, as we could see, they're different. So, we're going to multiply the first fraction by 7 over 7. And we're going to multiply the second fraction by the other do uh excuse me, the other denominator. So, that's going to be 5 over 5. And if you do this, you're going to get common denominators. So 7 * 2 is 14. 7 * 5 is 35. And then 4 * 5 is 20. And 7 * 5 is once again 35. So now that we have the same denominator, we can now add the numerators of the two ### Segment 2 (05:00 - 10:00) [5:00] fractions. 14 + 20 is 34. [snorts] And so the answer is 34 over 35. which corresponds to answer choice C. Number four, calculate the value of the expression shown below. So in this case, we need to use order of operations. Perhaps you heard of the expression PEMDAS. Please excuse my dear unsal. And it tells us the order in which we have to perform mathematical operations. P has the highest priority which stands for parenthesis. E is for exponents. M is for multiplication. D is for division. A is for addition. S is for subtraction. So what this tells us is that we need to work with the numbers inside of the parentheses first. 7 - 4 is 3. Next we can focus on the exponents. So what is 3 to the 2 power? power means that we are multiplying two 3es together and 3 * 3 is 9. 2 to the 4th power means that we're multiplying four 2's together. 2 * 2 is 4 * 2 is 8 * 2 is 16. So 2 to the 4th power is 16. Now at this point should we multiply or should we subtract first? Now according to the order of operations we need to multiply before we subtract. 5 * 9 is 45. 3 * 16 is 48. Now we can subtract. 45 - 48 is -3. And so we can write our answer in the box. And that's it. Number five. A circle has an area of 81 pi. What is its diameter? Well, let's begin by drawing a circle. And let's also draw the center of a circle. The distance between the center of the circle and any point on the circle is known as the radius of the circle. The diameter is basically twice the length of the radius of the circle. So the distance between any two points on a circle where those two points if you draw a line between them it passes through the center that distance is also the diameter. But what you need to know is this the diameter is just twice the value of the radius. So in order to calculate the diameter we need to calculate the radius of the circle first. Now we're given the area of the circle. What formula can we use? You need to know this formula. The area of a circle is equal to p * the square of the radius. Now we're given the area. It's 81 pi. So let's calculate r. Let's begin by dividing both sides by And so we could cancel pi. Thus we have 81 is equal to r^2. Now to calculate r we need to take the square root of both sides. 81 is 9. So now that we have the radius we can calculate the diameter. So the diameter is twice the value of the radius. It's going to be 2 * 9 which is 18. And so this is the answer. Answer choice E. Number six, which of the following is equivalent to the expression shown below? So here we need to subtract two polomials. Notice that we don't have a negative sign or any number in front of the first expression. So we really don't need to write the parenthesis. We can get rid of it. On the right side, we do have a negative sign in front of the second polomial expression. So we need to distribute that negative sign to these three terms. So it's going to be -5 x cub + 7x because we have two negative signs and then + 5. Now our next step is to combine like terms. These two are similar because they carry the term x cub with a coefficient in front. So we have 8 - 5 which is pos3. These two are not similar. They are not like terms. So we cannot add them. All we can do is just rewrite them. ### Segment 3 (10:00 - 15:00) [10:00] Now these two constants they are like terms. They're just numbers. 9 + 5 is -4. And so this is the answer. It's 3xqub + 5 x^2 + 7 x - 4. Number seven. What is the surface area of the cylinder shown below? For this problem, all you need is the formula. The surface area of a cylinder is equal to 2i r 2 + 2 rh. And you need to be able to identify the radius and the height of the cylinder. 7 is the radius of the cylinder. 12 is the height. So all we need to do in this example is replace r with 7 and then replace h with 12. [snorts] And then we just have to do the math. 7^2 is 7 * 7 which is 49. 7 * 12. 7 * 10 is 70. 7 * 2 is 14. If you add 70 and 14 that will give you 84. Now 2i * 49 is 98 pi and 2i * 84. 2 * 8 is 16. 2 * 4 is 8. So this gives us 168. And now we can add 98 and 168 together. And so that's going to give us 266 pi. For those of you who want to review the old school addition, you could also do it this way. 8 + 8 is 16. So you write the six, carry the 1. And then we have 1 + 6, which is 7. 7 + 9 is 16. Write the six, carry the 1, and then 1 + 1 is 2. And so you get 266 which means C is the right answer. Number eight, multiply. So how can we multiply these two fractions? One way of doing this is to multiply across. You can multiply 21 by 72 and 40 by 49. However, this is just a longer way of doing it because 21 * 72 is going to give you a big number and then you're going to have to simplify the fraction. But let's focus on a better way, a much easier way of doing this problem. It's best to break down these numbers into smaller numbers. 21, for instance, is 7 * 3. 40 is equivalent to 8 * 5. We can replace 72 with 8 * 9. And 49 is 7 * 7. So, make sure you know your multiplication tables. Now, you want to break it up in such a way that you could cancel a number on the top with a number that's on the bottom of the fractions. We can also cancel these two sevens. Now, once you've cancelled every possible number that you can cancel, now you want to multiply. 3 * 9 is 27. Okay, this computer's acting up. and then 5 * 7 is 35. So the answer is 27 over 35. And that's how you can multiply two fractions together. Break down the large numbers into small numbers and then cancel. Number nine, divide. So we have 54 over 48 / 63 over 35. Now the first thing that I would recommend doing is rewriting the problem like this. So we have the first fraction divided by the second fraction. Now when dividing two fractions together, you can change the problem into a multiplication problem. And there's something called keep change flip. And here's how you apply it. Keep the first fraction exactly the same as you see it here. and then flip the sign. Flip division to multiplication. I mean change the sign from division to multiplication and then flip the second fraction. So it's keep change flip. Keep the first fraction the same, change division to multiplication, flip the second fraction. So now let's do what we did before. 54. What two numbers should we break this down to? ### Segment 4 (15:00 - 20:00) [15:00] And what about 48? Look for numbers that can be found in each of these numbers. For instance, 54. You could say it's 27 * 2. It's 9 * 6. It's 18 * 3. But I will choose 9 * 6. The reason for that is because 48 is 8 * 6. And so I can cancel a six. 35 is 7 x 5. 63 is 7 x 9. So you always want to break it up in such a way that you can cancel the maximum number of numbers. So all we have left over is a five and an eight. So the final answer is 5 over 8. So that's how you could divide two fractions together. Number 10. If f ofx is equal to 2x^2 - 5x + 8, then what is the value of f of 4. So this problem is not too difficult. All we need to do is replace x with 4. So it's going to be 2 * 4^ 2 - 5 * 4 + 8. 4 2 is 16. 5 * 4 is 20. and 2 * 16 is 32. 32 - 20 is 12 and 12 + 8 is 20. So f of 4 is equal to 20. Which means that answer choice e is the answer. 11. What is the value of the expression shown below when a is equal to 4 and b is equal to5? So all we need to do here is we need to evaluate the expression. We just need to replace a with four and replace the variable b with5. After that it simply becomes an order of operations problem. So what's 4 to the 3 power? power is basically 3 4 multiplied to each other. 4 * 4 is 16 and 16 * 4 is 64. So 4 cubed is equal to 64. Now 3 * 4 is 12. Over here we have5^2 -5^2 is basically just two5s multiplied to each other. When you have a negative time a negative you're going to get a positive result. So this will be pos 255. So we're going to have 2 * 25us 2 * 4 is 8. Now our next step before we subtract 64 and 12, we really shouldn't do that. We should multiply -12 and5. Don't forget about pendas, which describes the order of operations. We want to multiply before we subtract. And if we multiply two negative numbers, it's going to give us a positive result. So 12 * 5 is 60. On the bottom we have 2 * 25 which is 50. Now 64 + 60 is 124 and 50 minus 8 is 42. But notice we could simplify this fraction. Because both numbers are even, we can divide the top and the bottom number by 2. So what is 124 / 2? Well 12id 2 is 6. 4 / 2 is 2. So that gives us 62 and 42 / 2. If we follow the same process, it's going to give us 21. And we can't simplify it any further than this. So this is our answer 12. What is the value of x in the equation shown below? So how can we solve this linear equation? Now the first thing that I would recommend doing is distributing the -2 to 3x + 4. and then combine like terms. So let's begin by doing that. -2 * 3x is -6x and -2 * 4 is8. Now everything else we're just going to rewrite it for now. By the way, for those of you who want to learn uh basic math, who want to improve your skills, let's say in fractions, addition, subtraction, things like that, or even algebra, when you get a chance, check the links in the description section below in this video. And I'm going to post my algebra ### Segment 5 (20:00 - 25:00) [20:00] playlists and some other videos on fractions and basic math so that you can boost your skills in this area if you need to do so. So let's get back to this. Our next step is to combine like terms. So these two are similar. 5x - 6x is - 1x. Now in order to solve an equation, your goal is to get the variable x on one side of the equation. All other I mean excuse me, all other numbers, you want to move it to the other side of the equation. So what I want to do is I want to add one x to both sides. By doing this all of the x variables will be on the right side because - 1 + 1 adds up to zero. So because the x variables are now on the right side. The non-x variables basically the constants I want to move to this side. So to get rid of -32 I'm going to add 32 to both sides. And so now I have -8 + 32 which is the same as 32 - 8. So that's 24. And then 3x + 1 x is 4x. Now I need to get rid of the four. The four is multiplied to the x. The opposite operation of multiplication is division. So I need to divide both sides by four. 24 / 4 is 6. So therefore, x is equal to 6. Which means that answer choice e is the answer. 13. Which of the following could be a value of y that satisfies the equation shown below. So how can we solve this equation? What we need to do is get rid of the square. And one of the best ways to do that is to take the square root of both sides. The square and the square root will cancel. So on the left side, all we're going to get is y minus one. Now what is the square root of 36? The of 36 is 6, but it can also be -6. And here's why. We know that 6 * 6 is 36, but also -6 * -6 is 36 as well. And so what we need to do at this point is write two separate equations. The first one y's I mean y - 1 is equal to pos 6. And for the second one y - 1 is equal to -6. Now let's add one to both sides. 6 + 1 is 7. So this is one possible answer. y can be seven. And if we add 1 to -6, -6 + 1 is5. So the other possible answer is y could be5. Now we don't see 7 as a solution, but five is one of the answers. So a is the right answer. Now let's check our work. So these are the two possible answers that we have 5 and 7. And the original problem was y - 1^2 is equal to 36. Now if you ever come across an equation where you don't know how to solve it, you can always guess and check. You can plug in the answer choices and see which one works. So let's say if we plugged in 7, you want to make sure that the left side is equal to the right side of the equation. So 7 - 1^2 doesn't equal to 36. 7 - 1 is 6. 6^2 is 36. And so that works. Now we could try the other answer. Let's replace y with5. -5 - 1 is -6. And we know that -6^ 2, which is -6 * -6, there's two of them. and that's equal to 36. And so that's how you can check your work. Now, what about this one? How can we solve for x in this equation? Now, if you don't know what to do, you can always try the guess and check method. So, let's see if x is equal to 3. Let's replace all of the x variables with three. And let's see if the left side is equal to the right side. 3^2 is 9. 5 * 3 is 15. And 9 + 15 is 24. Now 24 does not equal 36. So therefore, a cannot be the answer. ### Segment 6 (25:00 - 30:00) [25:00] Let's try b. Let's see if that's going to work. So let's replace x with9. -9 *9 or9^2 that's going to be pos 81 pos5 *9 is going to be - 455 [snorts] and 81 - 45 let's use the old school method of subtraction 1 - 5 is a negative number so we need to borrow a one from 8. So the 1 becomes 11. And then 11 - 5 is 6. 7 - 4 is 3. So 81 - 45 is 36. And uh that works. So we can see that b is definitely a value of x that will satisfy the equation shown below. But now how do we actually solve it? Let's say if we don't want to use the guess and check method. What techniques do we have available to us? The first thing you want to do is move the 36 to the other side. On the right side, it's positive 36. But when you move it to the left side, it becomes - 36. It simply changes sign. Now, notice that you have a quadratic equation in the form of ax^2 + bx + c is equal to zero. When you see that, you can solve this equation in one of two ways. You could factor it or you could use the quadratic formula. I guess you could solve it by completing the square, but that's rarely used in this instance. But let's focus on a factoring this equation. Here's what you need to do. You need to find two numbers that multiply to - 36 but add to 5. And so let's make a list. We have 1 and - 36. If you take - 36 / 2, you get the factors 2 and8. If we divide this by 3, 3 and -12 will be another possibility. And you could keep, you know, going in order. If we divide - 36 by 4, we get -9. Now notice that if we add four and 9, we get5, not positive 5. But if we reverse the signs, we get the two numbers that we're looking for. -4 * pos 9 is - 36, but -4 + 9 adds up to pos5. And so once you find the two numbers and also if this is a tromial where the leading coefficient is one you can simply write it this way you could write it as a product of two binomials it's going to be x - 4 * x + 9. And to check this if you foil these two binomials you should get the original expression. So if we multiply x by x, that's going to be x^2. x * 9 is 9 x -4 * x is -4x -4 * 9 is - 36. And then if we combine these two, 9x - 4x, it gives us 5x. And so that's how you can tell if you factored it correctly. So once we factor the quadratic equation, what is our next step? Our next step is to set each factor equal to zero. This is based on the zero product property because let's say this is equal to 8 and that's equal to 0. 0 * 8 will be 0. 0 * anything is zero. So if one of these factors is 0, the whole thing is zero. For the first equation, if we add four to both sides, we get the first possible answer. X is equal to 4. Now we see4 here but it's not the same as pos4. So e is not the answer. For the second one we need to subtract both sides by 9 and so we'll get x is equal to9. Now this answer we do have here and so that's why answer choice B is the correct answer. So now you know how to solve quadratic equations. Number 15. Factor completely. How can we factor this expression? Now, it's going to be similar to the previous problem, but with one additional step. When you have a problem and an algebraic ### Segment 7 (30:00 - 35:00) [30:00] expression that you wish to factor, one of the first things you need to look for is the GCF, the greatest common factor, if you can take that out. And it turns out that we can. Notice that every term has at least one x variable. So the GCF will contain an x variable. And notice that the coefficients of every term can be divided by 2. So the greatest common factor is 2x. Now what is 2x cub / 2x? 2 / 2 is 1. x cub / x. What you need to do is subtract the exponents. 3 - 1 is 2 and so you get x^2. Now what is -10 x^2 / 2x -10 / 2 is 5 x^ 2 power / x to the 1st power is going to be 2 - 1 which is 1. So it's 5x and -48x / 2x. The x variables will cancel and so it's -48 / 2 which is -4. And so that's how you can factor out the GCF in this problem. By the way, I want to mention something important. And for those of you who want more examples on factoring, go to YouTube and type in factoring and then my channel name, organic chemistry tutor. All of my factoring videos should come up and so you can look for the topic that you need help with. And the same is true for the rest of this video. If you find a question where you want more example problems on it, chances are I already have a video on YouTube covering that topic. So, just search that topic and then type in the name of this channel, Organic Chemistry Tutor, and it should come up. Also, you can check out my algebra playlist in the links in the description section below. And you can find a lot of other algebra videos that cover this topic and other topics that we're going to talk about in this video. So, take a look at that when you get a chance. Now, let's get back to this. Right now we have a tromial where the leading coefficient is 1. And so in order to factor this expression x^2 - 5x - 24 like before in the last problem we need to find two numbers that multiply to -4 but that add to5. And so let's make a list. Let's divide -4 by 1. If we divide it by 2 we'll get -12. If we divided by 3,8 4 -6. Now 2 + -12 doesn't gives us uh5 it gives us -10. However 3 +8 does add up to5. And so this is the pair of numbers that we need to factor the expression. It's going to be x + 3 * x - 8. And so that's basically it for this problem. And looking at the answer choices, we could see that it correlates to answer choice C. 2x * x + 3 * x - 8. Number 16. Solve the equation for x. Now, we do have a quadratic equation, but it's not in quadratic form. And here's how you can tell. Here we have an x^2, an x, and a constant. So, that's a quick way to tell if you have a quadratic equation. What we need to do first is subtract 7x from both sides. So, on the right side, this will cancel. On the left side, we're going to have 6 x^2 - 29x + 28 = z. So now it's in quadratic form. It's in ax^2 + bx + c format. So at this point, we could use the quadratic formula or we could try to factor it. Let's try to factor it for this example. So, what we're going to do this time, because the lead coefficient is not one, we're going to use a different method. We're going to multiply the lead coefficient by the constant term. And so 6 * 28 is going to give us 168. I'm going to put that on top here because I'm going to need space. Now ### Segment 8 (35:00 - 40:00) [35:00] once you get this number 168, you need to find two numbers that add up to -29 but that still multiply to 168. So what two numbers multiply to 168 but add to -29? Let's begin by dividing 168 by -2. This will give us84. Now these two numbers do not add to -29. So let's move on to the next one. 168 / -3, that's going to be -56. And if we divide it by -4, let's get rid of this. We're going to get -42. Now, 5 doesn't go into 168, but 6 does because remember 6 * 28 gave us 168. Now, if we divide 168 by -7, that's going to give us -4. And if we divide it by8, we're going to get -21. Now, notice that these two numbers, they multiply to 168, but they add to -29. So, those are the pair of numbers that we're looking for. And what we're going to do at this point is that we're going to replace the middle term -29x with -8x and -21x. Our next step is to factor by grouping. In order to do this, take out the GCF in the first two terms. The greatest common factor between 6x^2 and 8x is 2x because they both contain at least one x variable and 6 and8 are both divisible by 2. 6x^2 / 2x is 3x. 8x / 2x is -4. Now let's do the same thing for the last two terms. Let's take out the GCF which is going to be -7 because 21 and 28 are both divisible by 7. -21x / -7 is 3x pos 28 / -7 is -4. So notice that we have a common factor in these two terms and the common factor is 3x - 4. When you see that, that tells you that you're on the right track. So, let's factor out these two factors. So, if we take out 3x - 4 from this first term, we're going to be left with 2x. And if we take it out from the second term, we're going to be left with -7. So, what you see here on the outside, both of them will go into the second parenthesis. Now at this point we can separate each factor into two equations. So for the first one we need to add four to both sides. And so we're going to get 3x is equal to 4. Our next step is to separate the three from the x by dividing both sides by 3. And so we get the first answer x is equal to 4 over 3. Now for the second one, we need to add 7 to both sides and then we need to divide both sides by two. So we get that x is equal to 7 / 2. Now that's the second answer but it's not listed. This answer is listed x = 43. So the correct answer for this problem is answer choice D. Now let's talk about the other way in which we can get the same answer. So, I'm just going to write down the two answers that we have, 43 and 7 over2, because you need to be familiar with this method of getting the answers. Now, granted, sometimes you can just plug in the answers, guess and check, and that'll save you time, but for those of you who want to know how to use the quadratic formula to solve this equation, you can watch this part. for the rest you can skip over it. So recall that we said this is in quadratic form ax^2 + bx + c. So what ### Segment 9 (40:00 - 45:00) [40:00] you need to know is that a is equal to 6, b is = -29 and c is equal to 28. So now we could use this formula. X is equal tob plus or minus the square of b ^ 2 - 4 a c / 2 a. So b is -29. b ^2 that's going to be -29^2. and then -4 * a which is 6 and c is 28 all divided by 2 a or 2 * 6. So this is going to be positive 29 and then plus or minus square now -29^ squar that's going to be 841 and then 4 * 6 is 24 * 28 that's 672 / 2 * 6 which is 12. Now 841 - 672 that's 169. And so we have 29 plus or minus the of 169 / 12. Now the square root of 169 is 13. Now at this point you need to break it up into two parts using the plus or minus symbol. So we can have 29 + 13 / 12 or 29 - 13 / 12. So 29 + 13 that gives us 42. And for the other one, 29 - 13 is 16. Now, we can reduce both of these fractions. 42 over 12. We could say that 42 is 7 * 6 and 12 is 2 * 6 which allows us to cancel the 6 giving us the answer 7 / 2 which is one of the answers that we see here. Now for the other one we can say that 16 is 4 * 4 and 12 is 4 * 3. So getting rid of the four we can reduce that fraction to 43. [snorts] And so as you can see we can get the same answer using the quadratic formula. Number 17 divide. So we need to divide a tromial by a binomial. What is x^2 + 13x + 42 / x + 6? When you see a question like this, what you want to try to do first is you want to factor the tromial in the numerator. So how do we do that? Well, we need to find two numbers that multiply to 42 but add to 13. So what are those two numbers? 2 * 21 multiplies to 42. 42 is also 3 * 14, but it's also 6 * 7. and 6 * 7. Well, 6 + 7 adds up to 13. And so that's what we want to use here. So this is going to be x + 6 and x + 7. So notice that we can cancel the x + 6 factor. So the final answer is just x + 7. And that's the best way to do these problems if it's very easy to factor it. So answer choice B is the right answer. Try this one. So this problem is similar to the last one. However, the leading coefficient of the tromial is not one. So it's a lot harder to factor the numerator. You can do it, but it's just going to take more time. So for this problem, I'm going to show you another method and that is long division. ### Segment 10 (45:00 - 50:00) [45:00] Whatever is in the numerator should go on the inside. And the basic process that you're going to follow is you're going to divide, multiply, and then subtract. So, first we're going to divide 20x^2 by 4x. And so, that's going to be 5x. And then we're going to multiply. 5x * 4x is 20x^2. And then 5x * -3 is -15x. After that, subtract. So, you could perform the subtraction by changing the signs. So positive 20x^2 becomes - 20x^2 and -15x becomes pos5x after you distribute the negative sign. So these two will cancel and then we have -43 + 15. -43 + 15 is -28. Now we could simply bring down the 21 because there's nothing to subtract it by. 2 -28x / 4x is -7. Now multiply 4x * -7 that's -28x and -7 * -3 is positive 21. After that subtract so these two will cancel. Those two will cancel 21 - 21 is zero and so the remainder is zero. Thus the final answer is what we see on top. So 20 x^2 - 43x + 21 / 4x - 3 all of that is equal to 5x - 7 which means b is the answer 19. Simplify the expression that you see on the board. Now before we do this it might be wise to go over some properties of exponents. Let's say if we have x^2 * x cub. When you're multiplying two variables together, you can add the exponents. This is equivalent to x^ the 5th power. So you add 2 + 3 and you get 5. Now here's why this works. X^2 is basically two X variables multiplied to each other. X cub is basically 3 When you put them together, you have a total of five x variables being multiplied to each other. So whenever you see an exponent, it's basically repeat multiplication. Now what about raising one exponent to another? What is x^2 raised to the third power? In this case, you shouldn't add two and three. Instead, you should multiply the two exponents. It's going to be 2 * 3, which is 6. Here's why. So, the three tells us that there's three x² terms multiplied to each other. And every x² is basically two x variables multiplied to each other. So, this gives us a total of six x variables multiplied to each other. Now what about division? Let's say if we have x to the 7th power / x^ the 4th power. To get the answer we need to subtract. So it's 7 - 4 which is 3. Now here's how that works. X to the 7 basically means that we're multiplying seven x variables together. x to the 4th means that we're multiplying four x variables together. And so we can cancel four on top and four on the bottom, which leaves us with three on top. And so that's why it's x cubed. Now, even though you may have more x variables on the bottom, notice what's going to happen. Let's say if we have x cub / x to the 5ifth power. In this case, you take the top number three, subtract it by the bottom number, and this gives you a negative exponent. If you get a negative exponent, what you could do is take the x variable, move it to the bottom, and the negative exponent will become a positive exponent. So, the answer is 1 /x^2. Now, let's explain why that works. X cub is basically three x variables ### Segment 11 (50:00 - 54:00) [50:00] multiplied to each other. and x to the 5ifth power represents five x variables. We could cancel three on top with three on the bottom. So this leaves us with two x variables on the bottom, which means the final answer is 1 /x^2. So now you know what to do if you come across a negative exponent. So now that we reviewed the rules dealing with exponents, let's finish this problem. So we're dividing which means that we need to subtract the exponents. X^ the 9th power / X^ 4th power. Let's do this one step at a time. This is going to be X raised to the 9 - 4. So initially everything will be in the numerator after you divide two monomials together. Y 5th / Y 8th. So it's going to be the top exponent minus the one on the bottom. and then z to the4 / z^8 that's going to be z to the -4 minus8 now if you have two negatives number I mean let me say that again if you have two negative signs next to each other they will change to a positive sign now 9 - 4 is 5 since the exponent is positive we could leave it in the numerator 5 - 8 is -3. Now, the exponent is negative, which means that we need to move the y variable to the bottom of the fraction, but we'll do that later. And then -4 + 8. -4 + 8 is the same as 8 - 4 if you reverse the numbers. And 8 - 4 is 4. And so, this gives us z to the 4th power. Now, let's move the y variable to the bottom. So we could change the negative exponent into a positive one. So the final answer is x to the 5ifth power z to the 4th power / y^ the 3 power. And so that correlates to answer choice D. Number 20. Simplify. So what do you think we need to do here? There's two rules that we covered in the last problem that we're going to use. So if you recall, whenever you raise one exponent to another, you need to multiply. And if you multiply, let's say, two monomials together, you need to add the exponents. 3 + 4 is 7. So keep that in mind. The first thing we need to do is basically distribute the exponents to everything we see here. 3 is the same as 3 to the 1st power. So we have 3 to the first power raised to the second power. So we need to multiply the exponents. So let's distribute the two to every exponent that we see. So 1 * 2 is 2 and then we have 2 * 3 is 6 and then 2 * 4 is 8. Now this is also 2 to the 1st power. So let's distribute the three to these three exponents here. 3 * 1 is 3. 5 * 3 is 15. -4 * 3 is -12. Now, our next step is to multiply. But first, let's find out what 3 ^ 2 is. 3 ^2 is 3 * 3, which is 9. And 2 to the 3 power, that's 2 * 2, which is 8. and 9 * 8 is 72. Now a^ 6 * a 15 we need to add 6 + 15 which gives us 21. And then b^ 8 * b^ -12 we need to add 8 and -12 which gives us -4. But we're going to take b and move it to the bottom so that we can get a positive exponent. This gives us the final answer 72 a to the 21 b to the 4th on the bottom which correlates to answer choice B.