Grouping Factors: Rewriting 9 X 2 X 5 With Parentheses

Hey guys! Let's dive into a fundamental concept in mathematics: grouping factors using parentheses. This might sound a bit intimidating, but trust me, it's super straightforward once you get the hang of it. We're going to break down the expression $9 imes 2 imes 5$ and show you how to rewrite it in two different ways by strategically placing parentheses. This is a crucial skill for simplifying expressions and understanding the associative property of multiplication. So, buckle up, and let's get started!

Understanding the Associative Property

Before we jump into rewriting the expression, let's quickly touch on the associative property of multiplication. This property basically tells us that when we're multiplying three or more numbers, the way we group them doesn't change the final result. In other words, it doesn't matter which pair of numbers you multiply first; you'll still end up with the same answer. Think of it like this: if you have three friends, it doesn't matter which two you hang out with first – the final outcome of the hangout session (hopefully a good time!) remains the same.

This property is what allows us to use parentheses to group factors in different ways without altering the product. For example, $(a imes b) imes c$ is the same as $a imes (b imes c)$. This might seem like a small thing, but it's a powerful tool in simplifying complex calculations. It allows us to choose the easiest or most convenient grouping to solve a problem. For instance, if we have $2 imes 7 imes 5$, it might be easier to first multiply $2 imes 5$ to get 10, and then multiply by 7. This makes the calculation simpler than multiplying $2 imes 7$ first.

Understanding this property is essential for mastering more advanced mathematical concepts, such as algebraic manipulations and equation solving. So, make sure you grasp this concept well. Now, let’s apply this understanding to our expression and see how we can rewrite it using parentheses.

Rewriting the Expression: Method 1

Okay, let's tackle our expression: $9 imes 2 imes 5$. The first way we can group the factors is by putting parentheses around the first two numbers. This means we'll multiply 9 and 2 first, and then multiply the result by 5. Here's how it looks:

$(9 imes 2) imes 5$

So, what's $9 imes 2$? It's 18, right? Now we have:

$18 imes 5$

And $18 imes 5$ equals 90. So, this first way of grouping gives us a final result of 90. It’s crucial to perform the operation inside the parentheses first, as this dictates the order of operations. This is a fundamental rule in mathematics, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

This method demonstrates how we can choose to group the first two factors, perform the multiplication, and then multiply the result by the remaining factor. It’s a straightforward approach and perfectly valid according to the associative property. However, there’s another way we can group these factors, and that might even make the calculation a bit easier.

Rewriting the Expression: Method 2

Now, let's explore another way to group the factors in our expression $9 imes 2 imes 5$. Instead of grouping the first two numbers, we can group the last two. This means we'll multiply 2 and 5 first, and then multiply 9 by the result. Let's see how it looks:

$9 imes (2 imes 5)$

What's $2 imes 5$? It's 10, right? So now we have:

$9 imes 10$

And $9 imes 10$ is simply 90. Notice that we arrived at the same final answer (90) as before! This perfectly illustrates the associative property in action. By changing the grouping, we didn't change the final product.

This second method highlights the flexibility that the associative property gives us. Sometimes, grouping numbers in a certain way can make the multiplication easier. In this case, multiplying 2 and 5 first gave us 10, which is a very easy number to multiply by. This can save time and reduce the chances of making errors, especially when dealing with more complex calculations.

Comparing the Two Methods

So, we've successfully rewritten the expression $9 imes 2 imes 5$ in two different ways:

1. $(9 imes 2) imes 5 = 18 imes 5 = 90$
2. $9 imes (2 imes 5) = 9 imes 10 = 90$

Both methods gave us the same result, which reinforces the associative property of multiplication. The beauty of this property is that it allows you to choose the grouping that makes the calculation easiest for you. In this particular example, grouping 2 and 5 together might be slightly easier because it results in multiplying by 10, which is a mental math shortcut.

However, the key takeaway here isn't just about finding the easiest method for this specific problem. It's about understanding the underlying principle and how you can apply it to a wide range of mathematical scenarios. Being able to regroup factors can be incredibly useful in simplifying expressions, solving equations, and even in more advanced topics like algebra and calculus. So, make sure you’re comfortable with this concept and can apply it confidently.

Why is This Important?

You might be wondering,