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Law of Exponents - II
Now we can expand the laws of exponents a little bit further. Back in the arithmetic module, we learned about the distributive law. And really, the distributive law is one of the big ones, it’s really one of the big mathematical ideas. And of course, the distributive law says that P to the M plus or minus N, what we can do is just multiply the P separately times each one of those terms.
That is the Distributive Law. Multiplication distributes over addition and subtraction. As it turns out, division also distributes over addition and subtraction. Much in the same way, exponents distribute over multiplication and division. So if I have a times b to the n, or a divided by b to the n, I can distribute the exponent to each factor.
So a times b to the n equals a to the n times b to the n, a divided by b, that fraction to the n, equals a to the n divided by b to the n. So, we can distribute an exponent across multiplication or division. Here’s a very quick numerical example. Suppose we have 18 to the 8th, well we know that we could write 18 as a product, we could write it as its prime factorization.
And of course, the prime factorization of 18 is 2 times 3 squared. So, 18 to the 8th equals 2 times 3 to the 8th, while we can distribute that exponent to each one of those factors,. We’ll get it 2 to the 8th and then we’ll get a 3 squared to the 8th. And for 3 squared to the 8th, of course you’ll use the rule for the product of, for a power to a power, which means multiply the exponents.
And we’ll get 2 to the 8th times 3 to the 16th. So notice it’s very easy to go from the prime factorization to the, from, of the individual number, to the prime factorization of one of its powers. Here’s a practice problem. Pause the video and then we’ll talk about this. Okay, in the numerator, all we’re gonna do is multiply out that 4.
We’re just going to distribute it to each one of those terms. And for each one of those terms, we’re going to have a power to a power, which means multiply the exponents. So we’re gonna wind up with x to the 8th y to the 12th. Then we have the, we have to deal with the division. Well, x to the 8th divided by x to the 5th.
We subtract the exponents that’s going to be x cubed. Y to the 12th divided by y to the negative 5th. That’s going to be 12 minus negative 5 which is 12 plus 5 which is 17 and that’s why we get x cubed y to the 17th. It’s important to be aware of a very common and tempting trap, because it’s close to what is true.
So now we are going to talk about a trap. First of all it’s legal to distribute multiplication over addition and subtraction. That’s 100% legal. It’s legal to distribute exponents over multiplication and division. That’s 100% legal.
But it’s illegal to distribute an exponent over addition and subtraction. So that line, that’s just the distributive log, that’s 100% legal. That’s one of the fundamental patterns in mathematics. This is an also, it is also a version of the distributive law, we are distributing the exponent over multiplication and division, that’s also 100% legal. The thing that is illegal, is distributing the exponent over addition or subtraction.
That is always illegal. And in fact m plus or minus n to the p means that we’re taking that, what’s in the parenthesis, m plus or minus n, and multiplying it by itself p times. So these were variables we’d have to foil out several times. So you’ll never actually have to do that, but it’s just important to keep in mind that, that’s what it would be, not multiplying, not raising the individual terms of those powers.
And I’ll say this is a very tricky one, because even when you understand that this third line is illegal, the human brain’s inborn pattern matching software is tempted to make that mistake again, especially when you are under pressure. So you really need to know this cold so that even when you walk into the test and you’re stressed in the middle of the test you don’t accidentally fall into making this mistake again because it is a very tempting trap.
Again let’s look at all this with numbers. Here is just the ordinary distributive law with numbers. Multiplication distributing over addition. Here is the distributive law with powers. So that exponent distributing over multiplication and division, but it would be illegal if we had 8 plus or minus 5 to the 3rd.
That would not be 8 to the 3rd plus or minus 5 to the 3rd. And one way to see this is to just think let’s just take the subtraction case. If we look at 8 minus 5 to the 3rd, well what is that? Of course that is 3 to the 3rd which is 27. Whereas if we looked at something different, 8 cubed minus 5 cubed. Well 8 cubed as we’ve mentioned in other videos is 512.
5 cubed is 125 and if we subtract them we get 387. And those two are not equal. So in other words, we get two different numerical answers and that’s why we can’t set those things equal. We can do some legal math when sums are differences of power.
First we need to go back to that most impressive pattern, the Distributive Law. Now this is very tricky. When we read this equation from left to right, we say that we are distributing P. When we read this equation from right to left, we say that we are factoring out P. So distributing and factoring out are two sides of the same coin. It’s just a matter of whether we are reading this equation going from left to right or from right to left, but it’s the same fundamental pattern.
It’s also important to remember that any higher power of a base is divisible by any lower power of that same base. Thus, in the sum of a higher power and lower power of the same base. The greatest common factor of the two terms is the lower powner, power, and this can be factored out because the lower power is always a factor of a higher power.
So, for example, 17 to the 30th plus 17 to the 20th. Well, first of all, we know that 17 to the 30th has to be divisible by 17 to the 20th. We know one is a factor of the other. And so 17 to the 20th is the greatest common factor of these two terms. So I’m gonna factor that out.
17 to the 30th, I can write that as 17 to the 20th times 17 to the 10. By the, by the multiplication of powers law, I can write it that way. And, of course, 17 to the 20th, I can write that as 17 to the 20th times 1. I factor out 17 to the 20th, and I get 17 to the 20th times parentheses, 17 to the 10th plus 1. And that is a factored out form of those powers.
Obviously, the powers here are too large to simplify any of these resultant terms, but if the two powers in the sum are closer, sometimes such simplification is easy. So I will say pause the video and see if you can simplify this and then we’ll talk about this. Okay.
3 to the 32 minus 3 to the 28. 3 to the 28th is a factor of 3 to the 32nd. In fact, 3 to the 28th is the greatest common factor of these two terms. So we’re gonna express both of them as products involving 3 to the 28th. So 3 to the 32, we can write that as 3 to the 28th times 3 to the 4th. And of course 3 to the 28th, we can write that as 3 to the 28th times 1.
Factor out 3 to the 28, we get 3 to the 4th minus 1. Now 3 to the 4th, that’s something we can calculate. 3^4. So one way to think about that is if you have memorized 3 to the 4 is 81, also 3 to the 4th is 3 squared square. Well 3 squared is 9 and 9 squared is 81.
So that simplifies to 81. I do 80 min, 81 minus 1. That’s 80. And so this is 80 times 3 to the 28. We don’t often have to solve for something in the exponent, because this usually involves much more advanced ideas than are found on the test.
The test will expect us to know that if the bases are the same, if we have be to the s equals b to the t. Then it must mean that the exponents are equal. That will be very important in the lesson, equations with exponents, which we’ll get to much later in the module. In summary, exponents distribute over multiplication and division, and those are the patterns.
Exponents do not distribute over addition or subtraction. Those are very tempting mistake patterns and we can simplify the sum or difference of powers by factoring out the lower power. And finally, if we have bases are equal and we have a to the m equals a to the n we can equate the exponents.
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