Distributive property with variables

We're asked to apply the distributive property. And we have 1/2 times the expression 2a-6b+8. So, to figure this out, I've actually already copy and pasted this problem onto my scratch pad. I have it right over here. 1/2(2a-6b+8). So, lemme just rewrite it. So, I'm gonna take, and lemme color code it, too, just for fun, so it's going to be 1/2 times, give myself some space, 1/2(2a-6b), so, 2a-6b, minus six, lemme write it this way - 6b, and then we have plus eight. Plus, and I will do eight in this color. +8. And so, i just need to distribute the 1/2. If I multiplying 1/2 times this entire expression, that means I multiply 1/2 times each of these terms. So, I'm gonna multiply 1/2 times this, 1/2 times this, and 1/2 times that. So, 1/2 times 2a, so this is going to be 1/2 times 2a, times, lemme do it in that same color so you see where the 2a came from. 1/2(2a) minus, minus 1/2(6b). Minus 1/2(6b). Times 6b+1/2(8). 1/2(8). And so, what's this going to be? Well, let's see, I have 1/2(2a). 1/2(2) is just one, so you're just going to be left with A. And then you have minus 1/2(6b). Well, we could just think about what 1/2(6) is going to be. 1/2(6) is going to be three, and then you still are multiplying times B. So, it's gonna be 3b. And then we have plus 1/2(8). Half of eight is four. Or as you can say, eight halves is equal to four wholes. Alright, so this is going to be four. So, it's a-3b+4, a-3b+4. So, let's type that in. It's going to be a-3b+4. And notice, it's just literally half of each of these terms. Half of 2a is A, half of 6b is 3b, so we have minus 6b, so it's gonna be minus 3b, and then plus eight, instead of that, half of that plus four. So, let's check our answer. And we got it right. Let's do another one of these. So, let's say, so, they say apply the distributive property to factor out the greatest common factor. And here, we have 60m-40, so lemme get my scratch pad out again. So, I'm running out of space that way. So, we have, write it like this. We have 60, 60m-40. Minus 40. So, what is the greatest common factor of 60m and 40? Well, 10 might jump out at us. We might say, okay, look, you know? 60 is 10, so we could say this is the same thing as 10 times six and actually, and then, of course, you have the M there, so you could do this 10 times 6m. And then you could view, you could view this as 10 times four. But we, 10 still isn't the greatest common factor. You'll say, well, how do you know that? Well, because four and six still share a factor in common. They still share two. So, if you're actually factoring out the greatest common factor, what's left should not share a factor with each other. So, let me think even harder about what a greatest common factor of 60 and 40 is. Well, two times 10 is 20. So, you could actually factor out a 20. So, you have 20 and 30m. Sorry, 20 and 3m. And 40 could be factored out into 20 and, 20 and two. And now 3m and two, 3m and two share no common factors. So, you know that you have fully factored these two things out. Now, if you think this is something kind of a strange art that I just did, one way to think about greatest common factors, you say, okay, 60, you could literally do a prime factorization. You could say 60 is two times 30, which is two times 15, which is three times five. So, that's 60's prime factorization. Two times two times three times five. And then 40's prime factorization is two times 20, 20 is two times 10. 10 is two times five. So, that right over here, this is 40's prime factorization. And to get out the greatest common factor, you wanna get out as many common prime factors. So you have, here, you have two twos and a five. Here, you have two twos and a five. You can't go to three twos and a five 'cause there aren't three twos and a five over here. So, we have two twos and a five here. Two twos and a five here. So, two times two times five is going to be the greatest common factor. So, two times two times five, that's four times five. Four times five, that is 20. That's one way of kind of very systematically figuring out a greatest common factor. But anyway, now that we know that 20 is the greatest common factor, let's factor it out. So, this is going to be equal to 20 times, so 60m divided by 20, you're just going to be left with 3m. Just going to be left with 3m. And then minus, minus 40 divided by 20, you're just left with the two. Minus two. Minus two, so let's type that in. So, this is going to be 20 times, 20(3m-2). And once again, we feel good that we, literally, we did take out the greatest common factor because 3m and two, especially three and two are now relatively prime. Relatively prime just means they don't share any factors in common other than one.