Question

You and your friend are trying to solve this problem: (−4)×(57)×(−34)×(715) Your friend says the best strategy is to do each operation in order from left to right following the order of operations. You see that changing the order of the numbers being multiplied will make the problem simpler. What fact can you use to convince your friend that your method is a valid way to solve the problem?

Answer 1

You can convince your friend by using the commutative property of multiplication, which states that the order in which you multiply numbers does not affect the product. In mathematical terms, this means that for any two numbers $a$ and $b$, the equation $a\times b=b\times a$ holds true.

To demonstrate this in the context of the problem $(-4)\times (57)\times (-34)\times (715)$, you can rearrange the factors in a way that simplifies the calculations. Specifically, you might consider grouping the negative numbers together and the positive numbers together to make the multiplication process easier.

For example, you could rearrange and group like this:

$$(-4)\times (-34)\times (57)\times (715)$$

Calculating the product of the two negative numbers first:

$$(-4)\times (-34)=136$$

Now you have:

$$136\times (57)\times (715)$$

Now, you can multiply $136$ by $57$ first if that seems simpler, or continue multiplying in any order you prefer, knowing that the final product will be the same regardless of the order you choose to multiply the numbers.

Thus, by using the commutative property and showing that rearranging the multiplication doesn’t change the end result, you can effectively demonstrate to your friend that your method is just as valid as following the left-to-right order.