Welcome on the presentation on exponents. If I were to ask you what 2x3 is? Well I think at this point it should be pretty easy for you. That’s the same thing as 2+2+2 which equal 6. And you don’t have to do this you all know that 2x3 is equal to 6. So what we’re going to do now is we’re going to learn exponents which are the same thing that multiplication is to addition, exponents are to multiplications. And I’ll explain that in a second and I know that probably confuse you.
If I were to say what 2³ is, instead of 2+2+2 this is equal to 2x2x2 which equals 8 or if I were to say 3² that is equal to 3x3 and remember if it’s 3x2 would be 3+3. So this first 3x3 this equals 9 and 3+3 equals 6, and the reason why I’m doing this is because there’s always this temptation when you first learn exponents is to multiply. When I first learn, I thought 3^2 or 3² and I would be like oh that’s 6 but you also remember not as 3x3 which equals 9. Let’s do some more problems.
If I were to tell you (-4²), once again that’s the same thing as (-4)(-4). We don’t learn from the multiplying negative numbers that a negative times a negative is a positive and then as 4x4 well this equals (+16). You don’t have to write the positive I’m just doing that for that emphasis. If I were to ask you what (-4)^3 is, well that equals (-4)(-4)(-4). We know already that (-4)(-4) that this equals 16, (+16) and then we multiply that times (-4) and that equals (-64).
So something very interesting here to observe, when I took a negative number and we call this the base, when the base is negative and the case (-4), and I raised it into an even power I got a positive number right? (-4) to an even power is (+ 16) and when I took a negative number to an odd power to the three, I got a negative number and that make sense because every time you multiply by a negative number again its switches signs. I’ll show you the, I guess you call it the simplest example, (-1)^1 is equal to (-1) because that’s just (-1) times itself one time and if I said (-1)^2, well that’s (-1)(-1), will that equals (+1). But then if I said (-1)^3 oh once again that’s (-1)(-1)(-1). Well now this equals (-1)(-1) is (+1)(-1) is equals (-1) again and I could tell you (-1)^51 is, because 51 is odd we know that is equal to (-1). If it was a 50 then it would be (+1). Hope I didn’t confuse too much. Let’s do a couple more problems.
If I ask you what 5^3 is well that’s equal to 5x5x5 is equals 125. Similarly if I were to ask you at (-5)^3 is, that would be (-5)(-5)(-5) which should be (-125). Now one principle of exponents that might not seem completely intuitive to you at first is when I raise something to the zero power. So let’s say I had 2^0, it turns out that anything to the 0th power is equal to 1. So 2^0 is 1, 3^0 is equal to 1, (-900)^0 is equal to 1 and let me see if I can give you a little bit of intuition of why that is actually the case.
So if I were to ask you, let’s do 3^4 that equals 3x3x3x3 which equals 81, 3^3 is equal to 27 that’s 3x3x3, 3^2 is equal to 9, 3 ^1 is equal to 3. And now were going to say what’s 3^0 power. We’ll already know, I already told you the rule anything 0th power is equal to one but this would hopefully give you some intuition. When we went from the 4th power and the 3rd power, we divided by 3. Eighty-one divide it by 3 is 27 when you work from the 3rd power and 2nd power we divide it by 3. We’ll work from the 2nd power from 9 to 3 we divide it by 3 so it kind of makes logical sense when we go from the first power to the 0th power we’ll just divide it by 3 again. So 3 divided by 3 is 1, hopefully that gives you a little bit of intuitive sense. You might want to replay that and think about why that is, and there’s actually other aspect of exponents and why this would also make sense. Why something to the 0th power is equal to 1. But let’s just do some more problems in the time we have. I don’t want to get you too confuse.
So if I were to ask you 7² or 7x7 that’s 49. If I ask you (-6)^3 that equals (-6)(-6)(-6), (-6)(-6) is (+36) times (-6) and that equals what? It’s 180 and 36 that’s (-216) if my mental math is correct. You can actually multiply it out but if we did, I think you’re getting the point at this point. If we said oh and another thing if I told you 0^100 well that’s pretty easy, that’s zero times itself 100 times which still equal to zero. If I were to ask you 1^100 well it is equal to one right, you can multiply 1 by itself as many times as you want you’re still going to get one and remember if I had (-1)^1000. Well this is an even exponent so you’re still going to get one. It was (-1)^1001 and then it would become (-1). I think you remember why this is because when you multiply negative times itself and even number at times the negative cancel out and then if you multiply it by (-1) more time it becomes a negative number again. But let’s just do some normal problems. I just want to make sure you got the basic of exponents down.
If I were to tell you 8² that equals 8x8 which equals 64, if I were to tell you 25² that’s 25x25 which equals 625. Powers of 2 always very interesting, especially interesting and if you one day go into computer science, so 2^4 that’s 2x2x2x2. So 2x2 is 4, so its equal 16 and I did something very interesting here kind of on purpose. Notice that 2^4 is equal to 4x4 right, because we did 4x4 here. And I’m going to detail more on this later but I want to think about what that means, because 4 itself is the same thing as 2². So we learned just real fast and 2^4 is the same thing as 2²x2². So I’ll let you sit and think about that but other than that I think you have the general idea of how basic exponent work and I think you’re ready to try the level one exponent module. Have fun.
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