Hey guys! Ever wondered how the distributive property works and how it connects with the order of operations? It's a fundamental concept in mathematics, and today, we're going to break it down step by step. We'll use a specific example to illustrate how this property holds true when we follow the correct order of operations. So, grab your calculators and let's dive in!
What is the Distributive Property?
Before we jump into the example, let's quickly recap what the distributive property is all about. In simple terms, the distributive property allows you to multiply a single term by two or more terms inside a set of parentheses. It's like saying, "Hey, I'm going to share this multiplication with everyone inside the parentheses!"
Mathematically, it looks like this:
a( b + c) = a * b* + a * c*
Here, a is being distributed to both b and c. This property is super useful because it helps us simplify expressions and solve equations more easily. Understanding the distributive property is crucial for tackling more advanced math topics, so let's make sure we've got a solid grasp on it. You'll find it pops up everywhere from algebra to calculus, so paying attention now will definitely pay off later. Plus, it's not just about memorizing a rule; it's about understanding how numbers and operations interact, which is a much more powerful way to learn. Think of it as adding another tool to your mathematical toolbox—one that you'll reach for again and again. Now, let's see how this plays out with our example.
The Example: 4/5(4/3 + 1/3) = 4/5 ⋅ 4/3 + 4/5 ⋅ 1/3
Now, let's tackle the example we have:
4/5(4/3 + 1/3) = 4/5 ⋅ 4/3 + 4/5 ⋅ 1/3
We're going to use the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to demonstrate that both sides of this equation are equal. This will show us that the distributive property is indeed working as it should. We'll start by simplifying the left-hand side (LHS) and then move on to the right-hand side (RHS). By breaking it down step by step, we can clearly see how each operation contributes to the final result. This isn't just about getting the right answer; it's about understanding the process and why each step is necessary. Think of it like building a house: each brick needs to be placed in the correct order for the structure to stand strong. Similarly, in math, each operation has its place, and following the order ensures we arrive at the correct solution. So, let's put on our mathematical construction hats and start building!
Left-Hand Side (LHS) – Following PEMDAS
Let's start with the left-hand side (LHS) of the equation: 4/5(4/3 + 1/3). Remember PEMDAS? First up, we need to deal with the parentheses. Inside the parentheses, we have 4/3 + 1/3. Since these fractions have the same denominator, adding them is a breeze! We simply add the numerators and keep the denominator the same:
4/3 + 1/3 = (4 + 1) / 3 = 5/3
Now, our LHS looks like this: 4/5(5/3). The parentheses now indicate multiplication. So, we multiply 4/5 by 5/3. To multiply fractions, we multiply the numerators together and the denominators together:
4/5 * 5/3 = (4 * 5) / (5 * 3) = 20/15
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
20/15 = (20 ÷ 5) / (15 ÷ 5) = 4/3
So, the simplified LHS is 4/3. We've successfully navigated the parentheses and multiplication steps, and we've arrived at a clean, simplified fraction. Remember, each step is like a piece of a puzzle, and we're putting them together one by one to reveal the final picture. Now that we've conquered the left side, let's turn our attention to the right side and see if it matches up!
Right-Hand Side (RHS) – Applying the Distributive Property
Now, let's tackle the right-hand side (RHS) of the equation: 4/5 ⋅ 4/3 + 4/5 ⋅ 1/3. Here, we see the distributive property in action! We've distributed the 4/5 to both 4/3 and 1/3. According to PEMDAS, we need to perform the multiplications before we do the addition. So, let's start with the first multiplication:
4/5 * 4/3 = (4 * 4) / (5 * 3) = 16/15
Next, we'll do the second multiplication:
4/5 * 1/3 = (4 * 1) / (5 * 3) = 4/15
Now, our RHS looks like this: 16/15 + 4/15. We're ready to add these fractions together. Lucky for us, they have the same denominator, so we just add the numerators:
16/15 + 4/15 = (16 + 4) / 15 = 20/15
Just like before, we can simplify this fraction by dividing both the numerator and the denominator by 5:
20/15 = (20 ÷ 5) / (15 ÷ 5) = 4/3
Guess what? The simplified RHS is also 4/3! We've successfully navigated the multiplications and addition, and we've arrived at the same fraction as the LHS. This is a huge step in demonstrating that the distributive property holds true. It's like finding the missing piece of a treasure map and realizing that it leads to the same X marks the spot! We're not just crunching numbers here; we're verifying a fundamental mathematical principle. So, let's celebrate this small victory and move on to the grand finale!
LHS = RHS: Demonstrating the Distributive Property
We've done it! We've shown that both the left-hand side (LHS) and the right-hand side (RHS) of the equation simplify to the same value. Remember:
LHS: 4/5(4/3 + 1/3) = 4/3 RHS: 4/5 ⋅ 4/3 + 4/5 ⋅ 1/3 = 4/3
Since LHS = RHS, we have successfully demonstrated that the distributive property holds true in this case. This might seem like a small example, but it illustrates a powerful concept that applies to all kinds of mathematical expressions. The distributive property is like a versatile tool in our math toolbox, allowing us to manipulate expressions and solve problems in different ways. It's not just about getting the right answer; it's about understanding the why behind the math. By breaking down the problem step by step and following the order of operations, we've not only verified the property but also deepened our understanding of how it works. So, give yourselves a pat on the back! You've just conquered a fundamental mathematical concept, and you're well on your way to becoming math whizzes!
Conclusion
So, there you have it, guys! We've successfully used the order of operations to demonstrate that the distributive property holds. By simplifying both sides of the equation step-by-step, we showed that 4/5(4/3 + 1/3) is indeed equal to 4/5 ⋅ 4/3 + 4/5 ⋅ 1/3. This exercise not only reinforces our understanding of the distributive property but also highlights the importance of following the correct order of operations (PEMDAS). Remember, math isn't just about memorizing formulas; it's about understanding the logic and principles behind them. Keep practicing, keep exploring, and you'll continue to unlock the fascinating world of mathematics! You've now got a solid understanding of a key concept that will help you tackle more complex problems down the road. And that's something to be proud of! Keep up the great work, and remember, math is like a muscle—the more you use it, the stronger it gets.
