I've written over here on the board, or put on the sheet that we had before
that has the properties of addition and multiplication with positive and negative numbers:
commutative property has to do with order; the associative property has to do with changing the grouping and expressions;
and the distributive property is the property that involves both multiplication and addition.
We want to use these properties to simplify some algebraic expressions. So, let's look at our first example:
we want to simplify -6(3x). Now, you might think to yourself, well, the answer to this is going to be -18x
and you're correct. You do multiply the 6(3) = 18; and then, the answer is -18x.
The reason why that's true is because of the associative property for multiplication. I'm going to change the grouping,
so that I multiply -6 and 3, and then what I get times x. Now, -6(3) = -18, times x.
So, the reason why -6(3x) = -18x is because of the associative property for multiplication.
Here's our next example: let's simplify 2 + (3 + x).
This time, I'm going to apply the associative property for addition to combine the 2 and the 3. So, I have (2 + 3) + x.
So, I've changed the grouping from 3 and x to 2 and 3. Now, I'll add 2 + 3; I'll get 5 + x. So, I've simplified this expression now,
using the associative property for addition, rather than the one for multiplication. So, as you go through and you work these problems,
what you're going to say to yourself is, "-6(3x) is -18x." You won't always show the step, after awhile.
But what you should know is the step is there, and it's because of the associative property that -6(3x) = -18x.
Here are some more examples: let's multiply 2(6x - 3y).
This is subtraction in here, and I'm going to use the distributive property to distribute this 2 over this difference.
So, I'm going to multiply 2(6x); and from that, I'm going to subtract, there's my subtraction sign, 2(3y).
I've applied the distributive property there to distribute the 2 over this difference: 6x - 3y.
So, I multiply 2 times this and 2 times this. Now, when we state the distributive property, we only state it in terms of addition.
But it works for subtraction, as well, because subtraction is addition of the opposite.
So, since we've defined subtraction in terms of addition, the distributive property works for subtraction as well.
Now, when I multiply 2 times 6x, I would apply the associative property for multiplication to multiply the 2 and the 6.
The result will simply be 12x; minus, there's my subtraction sign, 2 times 3y will be 6y. So, there is an application of the distributive property,
where I distributed the two over this difference; this expression, 12x - 6y, is exactly the same as this expression algebraically.
Let's look at our next one: 5 (7 - 4y). Again, with the distributive property, I multiply 5(7), and then, 5(4y).
5(7) = 35, -5(4y) = 20y. So, there's the simplified version of this expression, or the expressions I get
when I apply the distributive property. Now, notice that 5(4y) = 20y. I don't distribute multiplication over multiplication.
That is, I'm not multiplying 5(4) and 5(y). I use the distributive property when, inside the parentheses, I have addition or subtraction.
Here's our next example; this one involves exponents.
And we're going to simply use our definition for exponents to expand each expression. This will be 4x(4x)(9xy)(9xy).
That's what these exponents 2 mean, is to write this down as a factor twice, and this as a factor twice.
Now, with the commutative and the associative properties in multiplication,
I'm going to change the order and the grouping of these numbers in this product, and write this as:
(4)(4)(9)(9)(x)(x)(x)(x), there's all four of the xs, times (y)(y).
So, that is simply an application of the commutative and associative properties for multiplication.
I group all the numbers together, here; and then, all the x's over here, and the y's over here.
Now, I'll continue to simplify. 4(4) =16, times (9)( 9). So, I'll end up there, if I do that multiplication, with let's see:
(16)(81) = 1296, when I multiply all this out, (x)(x)(x)(x) = x⁴,
and then, y(y) = y². So, when I simplify this expression,
I end up with 1,296x⁴y². That's the equivalent of that expression right here.
And I've done it by using my definition for exponents to just write these factors down twice. Then, I regroup;
I change the order and the grouping, using the commutative and associative property. So, I have the number part, the x's, and then the y.
And then, I just write it in a nice simplified form like that. Next, we want to combine similar terms. But before we do,
I want to go over and show you a definition I've written for similar terms: similar terms are two or more terms with the same variable part.
So for instance, 3x and -7x are similar terms because the variable part is the same, as are 4a and 5a.
Now, we can combine similar terms by using the distributive property in kind of a backwards way.
We add 4a + 5a by rewriting it as (4 + 5), that quantity, times a. So, in a way, we factored that a outside; so,
I have a(4 + 5); then, 4 + 5 we know is 9; and that's why 4a + 5a turns out to be 9a because of the distributive property.
The only kind of terms that we can combine are similar terms because I have to have the same variable part in both places
in order to apply the distributive property in this direction here.
So, let's go to the board now and look at a couple of examples: we want to combine 3x + 5x.
Okay, using the distributive property in the reverse direction from the way we've applied it before:
we write this as (3 + 5)x. Now, notice that I can simply multiply this out: x(5) and x(3) will be 3x + 5x.
So, I know that I haven't made a mistake, and this is an application of the distributive property.
But 3 + 5 = 8, and that's why I can write this as just 8(x). So, 3x + 5x = 8x. How about 6y - y?
Let me take away this set of parentheses right here. 6y - y is equal to, in parentheses, (6 - 1)y,
because the number of y's I have right here is 1y. And remember I have to write this so that if I apply the distributive property back the other way,
I get what I started with: y(1) = 1y; y(6) = 6y. So, that I know I've done this step correctly. Now, (6 - 1)y = 5y.
So, isn't that true that 6y - y would come out to be 5y?
So, I'm subtracting these numbers in front, which we call coefficients. 6y - 1y = 5y. Now you may skip this step right here,
as you get better at these problems, but you need to know the reason why 3x + 5x = 8x is because of the distributive property.
The distributive property is the thing that allows us to combine similar terms; and the only kind of terms that we can combine are similar terms.