Solving M * N = 12: Multiplication Properties Explained

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Hey guys! Let's dive into a fun math problem where we explore the magic of multiplication properties. We're given that m * n = 12, and our mission is to figure out how to solve two related expressions using some cool multiplication tricks. We'll break down each step so it’s super clear and easy to follow. So, grab your thinking caps, and let's get started!

Understanding Multiplication Properties

Before we jump into solving the expressions, let's quickly recap the main multiplication properties that will help us. Understanding these properties is crucial for tackling problems like these, so pay close attention!

  1. Commutative Property: This property tells us that the order in which we multiply numbers doesn't change the result. In other words, a * b = b * a. For example, 2 * 3 = 3 * 2. This is super handy because it lets us rearrange numbers to make calculations easier.

  2. Associative Property: This property allows us to group numbers in different ways when multiplying three or more numbers without changing the outcome. Mathematically, it means (a * b) * c = a * (b * c). Think of it as being able to choose which pair of numbers you want to multiply first. For instance, (2 * 3) * 4 = 2 * (3 * 4). This is incredibly useful when we want to simplify complex expressions.

  3. Identity Property: This one’s simple but powerful. It states that any number multiplied by 1 remains the same. So, a * 1 = a. The number 1 is the multiplicative identity, and it’s like a secret weapon for simplifying equations.

Knowing these properties will make solving our problem a breeze. We'll see how each one comes into play as we work through the expressions. So, let's keep these in mind as we move forward!

Solving (m * 8) * n = ?

Okay, let's tackle our first expression: (m * 8) * n = ?. Remember, we know that m * n = 12. The goal here is to use the properties of multiplication to rearrange and simplify this expression.

First, let's rewrite the expression using the associative property. This property allows us to regroup the numbers without changing the result. So, we can rewrite (m * 8) * n as m * (8 * n). See how we just shifted the parentheses? That’s the associative property in action!

Now, let's use the commutative property. This property lets us change the order of multiplication. We can rewrite m * (8 * n) as m * (n * 8). Why did we do this? Because we want to bring m and n together, since we know that m * n = 12.

Next, we can rewrite m * (n * 8) as (m * n) * 8. Again, we’re using the associative property, but this time in reverse. We’re grouping m and n together. This is a crucial step because now we can substitute the value we know: m * n = 12.

So, substituting 12 for m * n, we get 12 * 8. Now, it’s just a simple multiplication problem. What’s 12 times 8? It’s 96!

Therefore, (m * 8) * n = 96. We’ve solved it by strategically using the associative and commutative properties. Isn't it cool how we can manipulate expressions to make them easier to solve?

Here’s a quick recap of the steps we took:

  1. (m * 8) * n (Original expression)
  2. m * (8 * n) (Associative property)
  3. m * (n * 8) (Commutative property)
  4. (m * n) * 8 (Associative property)
  5. 12 * 8 (Substitute m * n = 12)
  6. 96 (Final answer)

Solving (m * 25) * (4 * n) = ?

Alright, let’s move on to our second expression: (m * 25) * (4 * n) = ?. This one looks a bit more complex, but don't worry, we'll use the same multiplication properties to break it down. Remember, our trusty fact is still m * n = 12. Let's see how we can use that!

First, let's rewrite the expression using the associative property. We can remove the parentheses and write it as m * 25 * 4 * n. This makes it easier to see all the elements we’re working with.

Now, let's bring the numbers together. We want to pair 25 and 4 because they multiply nicely to a round number. To do this, we'll use the commutative property to rearrange the terms. We can rewrite m * 25 * 4 * n as m * n * 25 * 4. See how we swapped the positions of n and 25? That's the commutative property at play.

Next, let’s use the associative property to regroup the terms. We’ll group m and n together, and 25 and 4 together. So, m * n * 25 * 4 becomes (m * n) * (25 * 4). This is a key step because we can now substitute the value of m * n.

We know that m * n = 12, so we can substitute that into our expression. This gives us 12 * (25 * 4). Now, let’s simplify (25 * 4). What’s 25 times 4? It’s 100! So, our expression becomes 12 * 100.

Finally, we just need to multiply 12 by 100. And that’s super easy, right? 12 times 100 is 1200. So, (m * 25) * (4 * n) = 1200. We did it! By applying the associative and commutative properties, we turned a complex expression into a simple calculation.

Here’s a summary of the steps we followed:

  1. (m * 25) * (4 * n) (Original expression)
  2. m * 25 * 4 * n (Associative property)
  3. m * n * 25 * 4 (Commutative property)
  4. (m * n) * (25 * 4) (Associative property)
  5. 12 * (25 * 4) (Substitute m * n = 12)
  6. 12 * 100 (Simplify 25 * 4)
  7. 1200 (Final answer)

Conclusion: Mastering Multiplication Properties

So, there you have it, guys! We’ve successfully solved both expressions by using the associative and commutative properties of multiplication. Remember, the key to tackling these types of problems is to understand the properties and apply them strategically. By rearranging and regrouping the terms, we can simplify complex expressions and make them much easier to solve.

Isn't it amazing how these properties work? They're like the secret ingredients in a recipe, helping us cook up the right answers every time. Keep practicing with these properties, and you'll become a master at manipulating and solving multiplication problems. Math can be fun, especially when you have the right tools and tricks up your sleeve! Keep exploring and keep learning! You've got this!