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Law of Exponents - I
In this video, we can start talking about the Laws of Exponents. Now these are really big. These are really important on the tests because these are patterns that are true for all numbers. The first situation concerns multiplying two powers. When have a to the n time a to the m.
Now, one thing that happens folks get stuff thinking about these things very abstractly. And they have, they, they force themselves to memorize a bunch of abstract rules. And then, it’s hard to remember which rule is which and they get themselves all confused. I’m going to urge you to think in terms of the fundamental definition of what an exponent is.
If you can go back to the fundamental definition of what an exponent is and understand the law from that perspective then you will really understand it. So, in order to understand multiplying two powers, let’s think about numbers. If I have 9 to the 5th times 9 to the third, well think about what that means. What fundamentally is 9 to the 5th? 9 to the 5th fundamentally means 9 factors of 5 multiplied together.
And nine cubed means three factors of nine multiplied together. So in other words I’m multiplying something with five factors of nine times something with three factors of nine. And of course when I multiply, I can just put all those factors together. And of course, what do I have there? How many factors?
Well I have 5 plus 3, I have 8 factors of 9, so of course that equals 9 to the 8th, so clearly 9 to the 5th times 9 cubed equals 9 to the 8th. Now, think about that a little bit more abstractly. We have seven to the m times seven to the n. The first contains m factors of 7 multiplied together, and the second contains n factors of 7 multiplied together.
So if I just multiply all the factors of seven together, what I’m gonna get is a whole string of factors of seven and there will be m plus n factors of seven in that string. And so what this means is that this should equal seven to the m plus n. Now we can treat this entirely in variables. If I have a to the m times a to the n, think about this again.
The first contains m factors of a. So I have a times a times a, m times. The second one contains n factors of a. So a times a times a n times. Multiply them all together, I’m going to get this long string of factors of a. And the number of factors of a in that string will be n plus m.
So in other words, we can just add those two numbers, that’s the new exponent for a. And that right there is one of our laws of exponents. Multiplying two powers of the same base, means that we can add the exponents. Now the next question concerns what happens when we divide power. Again, don’t think in terms of just an abstract law, let’s go back and think this through in terms of the fundamental definition of an exponent.
If you understand this law from the perspective of fundamentally what is an exponent, then you will really understand it. So I’m gonna suggest starting with numbers first. Let’s say 12 to the 7th divided by 12 Q. How would we figure this out? Well, 12 to the 7th has to be 7 factors of 12 multiplied together.
We have that in the numerator. In the denominator, we’re gonna have 3 factors of 12 multiplied together. So, we’re gonna have something like this. Well, obviously, we’re gonna get some cancellation here. We’re gonna be able to cancel 1, 2, 3 factors of 12 in the numerator and denominator.
So, everything in red there, all that cancels. And of course when you cancel, it just becomes one. So, it’s one times everything else. And so, we are just left with four factors of 12 and that’s 12 to the fourth. So, 12 to the seventh divided by 12 to the third equals 12 to the fourth. Right there, that suggests the rule.
We’ll make this a little more abstract. 12 to the m over 12 to the n. Well here we’re gonna have a fraction, and in the numerator of the fraction we have m factors of 12. In the denominator we have n factors of 12. And we’re gonna assume that m is greater than n, at least in this video.
This means that n factors of 12 will cancel. So all the factors in the denominator will cancel and that will remove some of the factors from the numerator and what will be left in the numerator after we remove those n factors that canceled, we’ll be left with m minus n factors of 12. And so what we’re gonna be left with is just 12 to the m minus n.
Now we can do that entirely in variables, think through it in variables. We have a to the m divided by a to the n. So this is a fraction, and the numerator we have m factors of a, in the denominator we have n factors. For this video, again, we’re gonna assume that m is greater than n.
So all those factors in the denominator will cancel, when we take the m factors in the numerator and remove the n factors that cancel. We’re gonna be left with m minus n. And that’s gonna be the exponent of a. And right there is our second law of exponents. When we divide powers of the same base, this means that what we have to do is subtract the exponents.
At this point, we can talk about a zero exponent. So far we’ve talked only about positive integers, but now we can expand to zero. What does a to the 0 mean? Using the division law we have just discussed, it wouldn’t be hard to create a zero exponent.
In other words, all we’d need is to divide two powers that have the same exponent and the subtraction will lead to zero. So for example, if I divide a cubed divided by a cubed, according the law of exponents that we’ve just had derived here. This means subtract the exponents. This should be a to the 3 minus 3, which is a to the 0.
But, of course, anything over itself must equal one. A to the 3 over a to the 3, that’s something over itself that has to equal 1. So a to the zero, equals one. Now is this law always true? Notice the only assumption we made in that argument was that the division was legal in the first place.
Obviously it wouldn’t be legal if a equals 0, but it would be legal for every other value. So we can say if a is unequal to 0, then a to the 0 equals 1. And as far as what happens with 0 to the 0, don’t worry about that. That’s something we get into in more advanced areas of mathematics. You do not need to worry about that for the test.
But you do need to know that for anything other than 0, raising it to the 0 power is equal to 1. The last scenario we will discuss is when we have a power to a power. So I have a to the m, and that whole thing to the n. And again, we’re gonna think this through in terms of the fundamental laws of exponents.
If you notice, in terms of the fundamental laws of exponents, you will understand it much more deeply. So, let’s start with numbers. 6 of the 5th, and that whole thing cubed. Well, of course, 6 of the 5th, what that means fundamentally is 5 factors of 6 multiplied together.
So, we have 5 factors of 6 multiplied together in the parentheses,. And we are cubing that. Well, what does it mean to cube something? To cube something, means that we multiply by itself 3 times. So we you take that parenthesis and it’s that parenthesis times itself 3 times. So that would be expanded out, what we mean with all the factors.
Well, how many factors do we have there? Well, we have 3 parenthesis in each one has 5 factors in it. So that means when we multiply it all together, we must have 3 times 5, or 15 factors of 6. And so, of course, it would be 6 to the fifteenth. Now we’ll do this a little more abstractly.
Inside the parentheses, we have m factors of a. When we raise this to the power of n, we have n different sets, each with m factors of a. In other words, we have a total of m times n factors of a. And so that means that the exponent has to be a to the m times n, and that’s our law of exponents.
Raising a power to a power results in multiplying the exponents. So far, the law of exponents we have reviewed here are, so product of 2 powers means add the exponents, quotient of 2 powers means subtract the exponents, a to the 0 equals 1. And power to a power means multiply the exponents. And once again, please do not memorize these in an abstract, rote fashion.
Please understand the arguments going back to the fundamental definition of what an exponent is. Understand those arguments and think through them and then you’ll really understand why they’re true. I also wanna mention some common mistakes with exponents.
It’s good to know, not only the patterns of what is true, but also the typical mistake patterns because the test always likes to test those particular mistake patterns. First of all, notice that all these laws work if the bases of the two powers involved are the same. We cannot apply any of these rules if the bases are different.
So for example, if we have things like this, we absolutely cannot apply the law of exponents. 2 cubed times 3 to the 5th does, that does not equal 6 to the something. If we have 12 to the 7th divided by 4 to the 5th, that does not equal 3 to the something. So there are no laws of exponents that are relevant if the bases are different.
Another mistake folks make when working quickly is to carry the operation on the powers to the exponent itself. In other words, when they’re multiplying the powers, people make the mistake of multiplying the exponents. Okay, that’s a very common mistake when people are working quickly and of course the correct thing to do is add the exponents.
Or, when people are dividing powers, they mistakenly divide the exponents. And again, the correct thing to do is to subtract the exponents. This is a really good one, raising a power to a power cuz there’s all kinds of mistakes people can make. So either, they mistakenly raise the first exponent of the second exponent. So that’s one kind of mistake people can make.
Another kind of mistake that people can make, is they confuse this with the product of big. The product of powers laws and they mistakenly add the exponents. So that’s another kind of mistake people can make. And of course what you’re supposed to do here, is you’re supposed to multiply the two exponents.
So 61 to the 14th, that would be the correct answer. Finally, there is no law for the sum or difference of powers. So if we’re adding three to the fourth plus three to the seventh or 5 to the eighth minus 5 squared. There is no fixed pattern for this. There is no single law of exponents.
In a couple lessons, we’ll learn how to use factoring out to simplify this kind of situation. So in other words, we may see this, but it’s not a simple law of exponents. It’s a somewhat more sophisticated trick that we’ll discuss in a couple videos. Overall, remember as in any branch of mathematics understanding means not knowing not only the rule itself, but also why it is true and what the common misunderstanding are.
So that’s a really deep understanding if you can understand from that perspective. In summary, we talked about these laws of exponents. We talked about some common exponent mistakes. And we talked about how there’s no exponent law for the sum or difference of powers.
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