Tutor profile: Sandy L.
What is the meaning of raising a number to the power of zero?
The first time we teach students about exponents we say that the exponent is the number of times that we multiply the base to itself. For example, 2^4 = 2x2x2x2 = 16 2^3 = 2x2x2 = 8 2^2 = 2x2 = 4 So what would the next power of 2 be? If we follow the pattern on the left, the next exponent would be what? (one). Ok, but it doesn’t make sense to say, “We multiply 2 to itself one time”, so there needs to be a new way to define 2^1 that doesn’t involve multiplication. So far we have... 2^4 = 2x2x2x2 = 16 2^3 = 2x2x2 = 8 2^2 = 2x2 = 4 2^1 = ? Take a moment to look at what’s happening down the right-hand side and see if you notice a pattern. Each number on the right is divided by 2 to get the next number. 16 divided by 2 is 8 8 divided by 2 is 4 4 divided by 2 is… 2. So we define 2^1 as 2. (Yes, everyone pretty much knows that, and it’s no big deal really, but we want to be mathematically precise here) Continuing on, so far we have... 2^4 = 2x2x2x2 = 16 2^3 = 2x2x2 = 8 2^2 = 2x2 = 4 2^1 = 2 2^0 = ? And 2 divided by 2 is… one. We want the pattern on the right to continue in the same way, so yes, 2^0 actually does equal 1 ! (and not zero, as many people might think). Another example, this time with the base of 3. 3^4 = 3x3x3x3 = 81 3^3 = 3x3x3 = 27 3^2 = 3x3 = 9 3^1 = 3 3^0 = ? Clearly we see the division pattern on the right, 81 divided by 3 is 27 27 divided by 3 is 9 9 divided by 3 is 3 3 divided by 3 is 1. So here we are again, at 1. Yes, 3^0 = 1 and 2^0 = 1 as well. In fact, no matter what base we use, we will always get to 1 at that last step. In general we say that a^0 = 1 for any base "a", as long as the base is a positive number. Oh, and one last note: It turns out that 0^0 does not have a mathematical meaning, so the base “a” must be strictly greater than 0 for our definitions to work. (The reason for this is a little involved, and best left to a discussion later in your mathematical journey). In our next lesson, negative exponents! Or perhaps you can figure out this on your own now that you know how to use the patterns in the lists. : )
Subject: Basic Math
Find the Least Common Multiple (LCM) of 12 and 18.
The Least Common Multiple (LCM) of two numbers is the smallest (or “least”) number that both numbers go into evenly. This turns out to be a very useful number to know when you need to add or subtract fractions. There are two main methods that are used to find the LCM, the “List Method”, and the “Exponents Method”. For this demonstration I’ll explain the List Method, first giving an explanation of what we mean by multiples, common multiples, and finally the least common multiple. Let's start by listing a few multiples of 12, and then of 18. These are the numbers 12x1, 12x2, 12x3, etc. Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, ... Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, ... (Clearly these lists go on forever, so we need to stop somewhere, and I wanted to list quite a few in order to explain how the List Method works. Don’t worry, though, when we actually get to finding the LCM, there will be a shorter way to do it.) Now let’s notice the numbers that these two lists have in common. Those numbers will be in bold (which unfortunately do no show up in this format): Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156,, 168, 180, ... Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, ... The common multiples of 12 and 18 on these lists are 36, 72, 108, 144, and 180. The smallest of these is 36, and so the LCM of 12 and 18 = 36. Now for the shortest way to find the LCM using a list. Notice that even though there are an infinite number of common multiples, since we are looking for the smallest common multiple we really don’t need to list so many multiples. We can stop as soon as we find the first match. Also notice that the list for the larger of the two numbers, 18 in this case, goes up in bigger leaps. In other words, the list for 18 hits those common multiples sooner than does the list for 12. So here is my method of finding the LCM using the List Method: Make a list of multiples only for the larger of the two numbers, and stop when you get to the multiple that the other number goes into. Like this… Multiples of 18: 18 (ask, “Does 12 go into 18?” No, so move on...), 36 (ask, “Does 12 go into 36?” Yes!). And you’re done. Another example: Find the LCM of 15 and 20. The larger number is 20, so we start there. Multiples of 20: 20 (ask, “Does 15 go into 20?” No.) 40 (ask, “Does 15 go into 40?” No.) 60 (ask, “Does 15 go into 60?” Yes!) The LCM of 15 and 20 = 60.
Find the slope of the line that passes through the points (4 , -1) and (10 , 11)
There are a couple of methods we can use to find the slope, either using a graph, or using a formula. We'll use the formula for this demonstration. The slope formula uses the numbers in the given points. (The formula has a lot of tricky symbols, but they are really just a way to identify the numbers in the points). Here is the official formula: (y2 - y1)/(x2 - x1). (Note: this looks a lot better on a Zoom whiteboard!) This formula is a way to calculate the number for the slope, which we also know as the ratio of the rise to the run when going from one point to another on the line. We usually say "rise over run" to describe this. Now back to the problem of finding the slope for the line through (4 , -1) and (10 , 11). The formula has the difference of the y-coordinates on top of the fraction, and the difference of the x-coordinates on the bottom. It doesn't matter which point you start with, but it's important to match the starting coordinates for the y's and x's. I'll show you what I mean by that as we use the formula. Our y-coordinates are -1 and 11, so we can write the top of the formula as (-1 - 11), which equals -12. (I'd be writing this on a whiteboard if we were actually in a tutoring session) Our x-coordinates are 4 and 10 ( Note: Since we started with the -1 for y, we have to start with the 4 for x), so the bottom of the formula will be (4 - 10), which equals -6. Put it together and we have -12/-6, which equals 2. So the slope of the line that passes through (4 , -1) and (10 , 11) is 2 . : ) But that's not the end of this story. One might ask, "What if we started with the other ordered pair first?", and that would be a very good question. The fancy formula with the fancy notation doesn't say which point is which. Lucky for us, it doesn't matter which point you start with, just as long as you keep the starting coordinates straight. So let's do this again where we start with the y-coordinate of 11 instead of the -1. (At this point we would do the problem all again from the start, but you, as the student, would walk me through it. You would see that the answer, instead of -12/-6 would be 12/6, which of course is still 2.) I just want to say that reading a math explanation is a lot different than hearing it in person, even if "in person" means online video format. (I'm sure you realize that, but I'm just sayin'.)