When we multiply positive and negative numbers, we simply multiply absolute values; regardless of the signs that we have,
we just multiply the absolute values. The answer to the product, the answer to the problem will be positive if both the numbers
that we are multiplying had the same sign, and the answer will be negative if the two numbers had different signs.
That's it for the rule for multiplication. Let's try some problems: 7(-6). Multiply absolute values: 7(6) = 42.
The two numbers have different signs, so the answer is negative. Next problem: -7(3); multiply absolute values; I get 21.
The two numbers have different signs, so the answer is negative. Now, you can see you can multiply two numbers with different signs,
just multiply the absolute values; the answer is negative if the two numbers had different signs.
Now, let's look at the problem in which the two numbers have the same sign: (-3)(-1); I multiply, 3(1) = 3.
The two numbers have the same signs so the answer is positive. The next problem I want to do is the product of three numbers.
We can only multiply two at a time though, so we'll just take them in order going left to right. So, -3(-4)(-5).
I'll multiply these two first: (-3)(-4) will be positive 12 because the two numbers have the same sign,
and that's times -5; 12(5) = 60, and the two numbers have different signs, so the answer is negative.
So, that rule for multiplication with positive and negative numbers is really pretty simple. We just multiply absolute values.
If the original two numbers have different signs, that answer is negative.
If the original two numbers have the same sign, the answer is positive.
Here's our next problem: let's simplify. This involves more than one operation. I'll work inside the parentheses first;
inside here, and inside here; 4 - 7 will be -3; 6 - 9 will be -3.
Now, when I multiply (-3)(-3), I multiply absolute values; I get 9. The two numbers had the same sign, so the answer is positive.
So, when I do this subtraction up here, I'm assuming that you've gotten to the point where you can do subtraction very easily.
And, when I say 4 - 7, you're thinking 4 + (-7), which is -3; 6 + (-9) = -3, again.
Now, the only thing that's new right here, really, is this multiplication: (-3)(-3) = 9.
Next problem: -4(3 - 8) - 6(2 - 5).
Now, this might look like the distributive property here, but it's not because these are just simply numbers that I can actually subtract. So,
-4(3 - 8) is the same as 3 + (-8), which is -5 - 6. 2 - 5 = 2 + (-5), which is -3.
Now, I have subtraction and multiplication. I have to multiply first: -4(-5) will be positive 20, here's my subtraction sign,
6(-3) = 18. Now, I could've thought of this as addition of the opposite and said plus (-6)(-3),
and that would have given me a positive 18. Either way, I'll get the same result.
Subtracting -18 is the same as adding positive 18, so I end up with an answer of 38.
So, I simplify according to the rule for order of operation: inside the parentheses, first. Then, I multiply.
Finally, the last thing is to subtract by adding the opposite. One more problem: let's simplify this expression; it has a variable.
This time, we have to apply the distributive property: -3(2x), subtract, -3(5), and then subtract just 7.
So, the parentheses end right here, so that's as far as I'm going to distribute that -3.
-3(2x), by the associative property, will be -6x because I'll multiply -3 and positive 2 to get -6;
subtract, there's my subtraction sign, -3(5) = -15; subtract 7;
-6x - (-15) is the same as adding positive 15. And, now, let's see,
I can add 15 + (-7), and I get -6x + 15 - 7, 15 - 7 = 8.
The only other part to this problem that could have been different would be down here, when I multiply (-3)(5),
I could have thought of this as adding -5 and multiply (-3)(-15), and gotten positive 15.
You see, that's the same result that I eventually ended up with down here.
So, that's a quick look at multiplication with positive and negative numbers.