I'm sure by now you have heard the expression, 'Please Excuse My Dear Aunt Sally' to describe the rules for order of operations. This silly mnemonic was created to help us remember to do parentheses first, followed by exponents, multiplication and division from left to right, and finally addition and subtractions from left to right.
In this lesson, it will be most important for us to look at the first part of our mnemonic, the parentheses. We know that whenever we see a set of parentheses in a problem, it screams, 'DO ME FIRST.' Therefore, it is necessary to note that we can add parentheses around any part of a problem to tell people that it should be completed first regardless of the operations inside of them.
Now, you are probably wondering why we just talked about order of operations when this lesson is supposed to be on the associative property of multiplication. But bear with me and I promise you will see in the next sections why it is important for us to have a clear understanding of parentheses to truly comprehend the use of the associative property of multiplication.
The associative property of multiplication states that when performing a multiplication problem with more than two numbers, it does not matter which numbers you multiply first.
In other words, (a x b) x c = a x (b x c).
So, no matter where we put our set of parentheses, we will still get the same answer. Now do you see why we talked about parentheses at the beginning of this lesson?
Let me guess. You are probably starting to wonder why you even need to put parentheses in the problem, right? Why not just work the multiplication from left to right? What is the point of adding yet another property to remember?
You might not believe me just yet, but the answer is simple. The associative property of multiplication makes multiplying longer strings of numbers easier than just doing the multiplication as is.
Let's look at the multiplication problem: 6 x 4 x 5
Solution #1: Doing the problem from left to right we would start by multiplying 6 x 4 = 24 and then multiplying the 24 by 5 to get a final answer of 120.
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If you are anything like me, it probably took a minute to do the math for 24 x 5 because it is not a simple multiplication problem and requires a little more thought.
Now, let's look at the same problem using the associative property of multiplication. Remember, the associative property just means that we can add parentheses around any two numbers to regroup what we multiply first.
Solution #2: Using the associative property, I am going to regroup the problem so that I multiply 4 x 5 first. Our new problem would then be 6 x (4 x 5).
Look closely; we have not changed anything about the numbers or the operations. We simply added parentheses around 4 x 5 to say that this is what we are going to do first.
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Now, doing the problem, 6 x (4 x 5), we are going to start with 4 x 5 = 20 because of our parentheses and then multiply 20 by 6 to get our final answer of 120.
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In both solutions, we got the exact same answer of 120. This proves that our associative property of multiplication works!
So why then did I choose to multiply 4 x 5 first instead of just working the problem from left to right? Look back at our first solution. In this case, it took a little extra work to multiply 24 x 5. However, by doing 4 x 5 first, as in our second solution, we made it much simpler to then multiply 20 x 6 to reach our final answer of 120.
There are two things that are important to understand when using this property.
The first is our rule of zeros. When you multiply a number by another number that ends in zero(s), in the case of our example, 20 x 6, you can drop off the zero(s), do the multiplication, and then add the zero(s) back on. So for 20 x 6 we would drop the zero making the problem 2 x 6 = 12 and then add the zero back to get an answer of 120. Quick and easy mental math!
It is also important to understand that while this property will also work for addition, it will NOT work for subtraction or division.
In this lesson, we learned that the associative property of multiplication is just a fancy way of saying that it does not matter which two numbers you multiply first in a multi-step problem. Whether you multiply from left to right or chose another grouping that simplifies the amount of work you do, you should ALWAYS get the same answer. So then, why not use this property and make your math work just a little bit easier?
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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons
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