In this video, we’re gonna be learning about the distributive property of multiplication, which is sometimes known as the distributive law. And we’ll be seeing a few examples of it in action.
First, let’s think about multiplication as repeated addition and visualize this as rows of objects. Here are four candles. And that’s one row of four candles, so that’s one times four. Now if I add another row of four candles, that’s two rows of four candles. So that’s two times four candles. And if I had a third row, I’ve now got three times four candles. Now if I add two candles to each row, I’ve got three lots of four plus two candles. And I could write this as three times four plus two or, as we would normally write it, three right up against the parentheses four plus two.
But another way of looking at this would be to think I’ve got three times four candles and three times two candles. So I’m distributing the three into the parentheses: three times four and three times two. So the distributive property means that when you see expressions like this, you can write them like this; three up against the parentheses four plus two means three times four plus three times two. Now obviously, we could interpret this as four plus two is six, so this is three times six. And we’ve got three times six, we get the same answer at the end of the day eighteen candles, but that’s not the distributive property. The distributive property is all about saying that three on the outside is distributed to each term inside the parentheses.
So the distributive property of multiplication over addition is that when I’ve got 𝑎 times 𝑏 plus 𝑐, it means I do 𝑎 times 𝑏, and then I add 𝑎 times 𝑐. And the distributive property of multiplication over subtraction says that if I’ve got 𝑎 outside the parentheses and 𝑏 minus 𝑐 inside, that equates to 𝑎 times 𝑏 minus 𝑎 times 𝑐. So the distributive property or the distributive law of multiplication works over addition and subtraction.
Okay let’s see a few examples with numbers in them and see how they work. Rewrite the expression five times nine plus seven using the distributive property. Well five times nine plus seven means five times nine plus five times seven. Well in fact that’s it, mission accomplished. We have just rewritten it using the distributive property. The question didn’t ask us to evaluate it or simplify it, it just said rewrite it using the distributive property. And an alternative way of expressing that answer is to leave the nine and seven in their own parentheses like this.
And here’s another question. Rewrite the expression three times eight minus two using the distributive property. So three times eight minus two means three times eight minus three times two. So we can write that three times eight minus three times two, and that would be our answer. Again, we’ve just rewritten the expression using the distributive property. We haven’t evaluated or simplified it. And again, we could leave the eight and the two in their individual parentheses. So we’ve got three times eight minus three times two written that way.
Now, rewrite the expression 𝑥 times seven minus four using the distributive property. And although this has got a letter in it, 𝑥, so it involves a bit of algebra. The principle is just the same as it was before. It means 𝑥 times seven minus 𝑥 times four. So that’s 𝑥 times seven minus 𝑥 times four. But with algebra, we tend to write the number first and the letter second, which would give us seven 𝑥 minus four 𝑥.
Now, rewrite the expression eleven times 𝑥 minus five using the distributive property. This time, the letter is inside the parentheses, but the process is just the same: eleven times 𝑥 minus eleven times five. And again just tidying up the algebraic expression, that leaves us with the eleven 𝑥 minus eleven times five. We could get on to simplify that to eleven 𝑥 minus fifty-five, but remember the question didn’t ask us to fully simplify. It just asked to rewrite the expression, so that’s enough.
And now we’ve got a sort of backwards distributive property question. Rewrite the expression seven times fifteen plus seven times eleven using the distributive property. So we’ve got a common factor of seven. It’s seven times fifteen, and then we’re adding seven times eleven. So we can write that as seven times fifteen add seven times eleven. It’s as simple as that. That’s our answer. Now in this question, we’re asked to evaluate the expression six times nine minus seven using the distributive property. So distributing the six across the parentheses is gonna be six times nine minus six times seven.
And six times nine is fifty-four and six times seven is forty-two. So we’ve got fifty-four minus forty-two, which is twelve.
Now the alternative way of evaluating that would have been to evaluate the contents of the parentheses first, so nine minus seven is two. So that becomes six times two, which again is twelve. Now that was quicker and it did give us the right answer. However, the question told us to use the distributive property, so we had to do that to get full marks.
So just to summarize what we’ve learned then, the distributive property or the distributive law of multiplication works over addition and subtraction. And its general form, it’s 𝑎 times 𝑏 plus 𝑐 means 𝑎 times 𝑏 plus 𝑎 times 𝑐. And 𝑎 times 𝑏 minus 𝑐 means 𝑎 times 𝑏 minus 𝑎 times 𝑐. And these are gonna obviously be written as 𝑎𝑏 plus 𝑎𝑐 and 𝑎𝑏 minus 𝑎𝑐.
Just one quick warning before we go. Be aware of signs. If we see something like negative three times two plus five, it’s negative three times two add negative three times five. And negative three times two is negative six and negative three times five is negative fifteen. So we’re adding negative fifteen, which means were actually taking away fifteen. And six times negative seven minus two is negative seven here, so this is six times negative seven take away six times two. And six times negative seven is negative forty-two, and we’re taking away six times twelve. So we’re taking away twelve. And let’s look at this extreme example: negative five times negative two take away negative four. So that’s negative five times negative two take away negative five times negative four. In fact, I’d probably put a little bit of parentheses around the negative five as well just so we know that that negative sign is associated with the five. And negative five times negative two, negative times a negative makes positive, so that’s positive ten. And negative five times negative four, well negative times negative makes positive, so five times four is twenty. So that’s positive twenty. And we’re taking away that positive twenty, so that means ten take away twenty.