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Video: Multiplication of Integers Using the Commutative Property

Tim Burnham

Using the properties of multiplication of integers, calculate 8×15×125.

03:22

Video Transcript

Using the properties of multiplication of integers, calculate eight times fifteen times one hundred and twenty-five. Okay, first let’s think about what the properties of multiplication of integers are.

The first property is closure: when you multiply two integers together, the result is always another integer.

Next, we’ve got commutativity. And that means it doesn’t matter what order you multiply your integers together in; you’ll always get the same answer. So if I’ve got two integers 𝑎 and 𝑏, whether I do 𝑎 times 𝑏 or 𝑏 times 𝑎 I’m still gonna get the same answer.

Thirdly, we’ve got associativity. And that means if I’ve got three integers that I’m multiplying together, it doesn’t matter if I multiply the first two together first and take that result and multiply it by the third integer or whether I take the second two integers and multiply those together and take that result and multiply it by the first integer. I’m still gonna get the same answer.

Fourthly, integer multiplication is distributive. And that means if I’ve got one integer times the sum of two other integers like 𝑎 times 𝑏 plus 𝑐 then that can also be written as 𝑎 times 𝑏 plus 𝑎 times 𝑐.

And fifthly, we have the property of identity. And that’s basically, if I multiply any integer by one I just get the same integer as an answer; and if I multiply any integer by zero, I just get zero as an answer.

So we’ve gotta use these properties in order to make the calculation of this a little bit easier. At first, I’m gonna take a look at these numbers: eight, five, and a hundred and twenty-five.

Well I know that four times twenty-five is one hundred, so eight times twenty-five is gonna be two hundred. And I know that eight times one hundred is eight hundred, so eight times one hundred and twenty-five is eight hundred plus two hundred, which is a thousand. So I’ve then got fifteen times a thousand, so it looks like that’s gonna be a good approach.

So my first step is gonna be to rearrange that calculation. It doesn’t matter what order I multiply them together in; I’m gonna get the same answer. So I’m gonna swap the fifteen and the hundred and twenty-five so that I can then tackle that eight times a hundred and twenty-five. So that step is the commutative property in action.

Next, I’m gonna decide to multiply eight times twenty-five before multiplying by fifteen, and that’s the associative property.

And then in preparation for the distributive step, I’m gonna split up a hundred and twenty-five into a hundred plus twenty-five. Now the distributive property tells me that’s the same as eight times a hundred plus eight times twenty-five.

And now I know that eight times a hundred is eight hundred and eight times twenty-five is two hundred. And of course, eight hundred plus two hundred is a thousand.

And lastly, a thousand times fifteen is fifteen thousand. So the answer is fifteen thousand; but because the question told us to use the properties of multiplication of integers, it was very important that I wrote the fact that we had this step being commutative, this step being associative, and this step being distributive.