Distributive Property: Equivalent Expression Explained

Hey math enthusiasts! Let's dive into a common problem: applying the distributive property to find an equivalent expression. The question at hand is: "Which correctly applies the distributive property to show an equivalent expression to $(-2.1)(3.4)$?" Don't worry, we'll break it down step by step, so you'll be acing these problems in no time. Understanding the distributive property is like having a superpower in algebra – it simplifies complex expressions and helps you unlock the secrets of equations. It's a fundamental concept, and once you grasp it, you'll find it popping up everywhere in your math journey. So, grab your pencils, and let's get started. We'll explore how to break down the original expression using this powerful property.

First, let's understand the core concept. The distributive property allows us to multiply a number by a sum or difference. The general form is a(b + c) = ab + ac. In our case, we're dealing with a product, and we want to expand it using this property. This means we'll be breaking down one of the factors into a sum or difference. The key is to recognize how to decompose -2.1 and 3.4 to make the multiplication easier. The goal is to rewrite the original expression in a way that is mathematically equivalent but more manageable. Think of it as a mathematical puzzle – we're rearranging the pieces while keeping the overall value the same. This ability to manipulate expressions is crucial for solving equations and understanding more advanced mathematical concepts. It builds a solid foundation for tackling more complex problems later on. So, remember that understanding the distributive property isn't just about memorizing a rule; it's about developing a skill that will serve you well throughout your mathematical journey. Let's start with the original expression: $(-2.1)(3.4)$. We need to look at each option provided and determine which one correctly applies the distributive property.

Breaking Down the Distributive Property

To correctly apply the distributive property, we need to expand the expression $(-2.1)(3.4)$. Let's analyze each of the multiple-choice options provided, one by one. The goal is to see which option correctly applies the distributive property to find an equivalent expression. Remember, the distributive property helps us to rewrite a product by multiplying each part inside the parentheses. In this context, we need to choose the one that expands the expression correctly. This process isn't just about finding the right answer; it's also about understanding why the other options are incorrect. This deeper understanding will solidify your knowledge and make you more confident when facing similar problems in the future. We're not just looking for an answer; we're seeking to understand the underlying principles. That way, you'll be able to solve similar problems with ease. This active learning approach is a great way to improve your skills. Now, let's examine the options and see which one does the job properly. Remember that the correct application of the distributive property must result in an equivalent expression to the original one.

Option A: $(2)(3)-(0.1)(0.4)$

Let's break down option A: $(2)(3)-(0.1)(0.4)$. This option attempts to use the distributive property, but the original expression is $(-2.1)(3.4)$. The numbers aren't correctly represented here, and the signs are off. When applying the distributive property, we need to ensure that the factors are accurately represented. It looks like it is trying to break down the original numbers, but it misses the negative signs, so this can't be correct. It seems like the negative sign on 2.1 is missing, and the product of (0.1)(0.4) is not part of the correct breakdown when applying the distributive property correctly. The whole expression is not equivalent to the original, so this option is incorrect. Remember, the distributive property requires a precise expansion of the original expression to maintain its equivalence. Double-check all signs and numbers when evaluating. The crucial thing to remember is the distributive property must accurately represent the factors in the original expression. Therefore, we can immediately cross out this option.

Option B: $(-2)(3)+(-0.1)(0.4)$

Now, let's consider option B: $(-2)(3)+(-0.1)(0.4)$. This is another attempt to apply the distributive property, but we can see some issues here. Although it correctly considers the negative sign in -2.1, it doesn't correctly apply the distribution across both parts of the original numbers. When you apply the distributive property, ensure each term in the expression accurately reflects the initial factors. It seems to have correctly used the negative on the 2, but not correctly. If we were to correct option B, it would be (-2)(3.4) + (-0.1)(3.4). Without this correction, option B is incorrect. This option is not correctly expanding the original expression and therefore will not give the same result. The negative sign is a key element and must be carefully addressed. So, again, this option is incorrect. To properly use the distributive property, we must break down one of the factors into a sum or difference and then multiply each term by the other factor, so this is not correct.

Option C: $(2.1)(3)-(2.1)(0.4)$

Alright, let's evaluate option C: $(2.1)(3)-(2.1)(0.4)$. This option is almost correct but is missing one crucial element – the initial negative sign. Remember, the original expression is (-2.1)(3.4). Option C has to account for the negative sign. It has applied the distributive property by separating 3.4 into (3 - 0.4), but the minus sign is not correct. When evaluating, it's essential to double-check that all signs are correctly represented. If the original number has a negative sign, that must also be included in the breakdown. Without that, the answer will not be the same. The lack of this makes this option incorrect as well. To make this correct, the expression must have a negative sign somewhere. Even though it breaks down the 3.4 into a difference, without the negative sign, it is incorrect. Let's see if the next option works.

Option D: $(-2.1)(3)+(-2.1)(0.4)$

Finally, let's analyze option D: $(-2.1)(3)+(-2.1)(0.4)$. This looks like the winner, guys! This option is the correct application of the distributive property. It correctly breaks down the 3.4 into (3 + 0.4) and distributes the -2.1 across both terms. So, let's break it down: (-2.1)(3.4) can be rewritten as (-2.1)(3 + 0.4). Applying the distributive property, we get (-2.1)(3) + (-2.1)(0.4). This is exactly what option D provides! It accurately reflects the original expression while properly distributing the multiplication. This option is the most valid way to apply the distributive property to the original problem. This means that if we were to calculate both expressions, the result would be the same. The use of the distributive property here provides an equivalent expression to the original one. Therefore, option D is the correct answer. This shows that the distributive property is not about memorization but about how to decompose and manipulate expressions.

Conclusion: Mastering the Distributive Property

So, there you have it, guys! The correct application of the distributive property to find an equivalent expression for $(-2.1)(3.4)$ is option D: $(-2.1)(3)+(-2.1)(0.4)$. Remember, understanding the distributive property is fundamental to algebra. It helps you simplify expressions and solve equations, so you'll encounter it repeatedly. By breaking down the expression and carefully considering each step, you can confidently tackle these types of problems. Keep practicing and remember the key to success: understanding the distributive property is not about memorizing a formula; it's about developing a skill that lets you manipulate and simplify mathematical expressions. That ability is essential in mathematics. So, keep up the great work. You guys got this! Remember to always double-check the signs and the numbers, and you'll be acing these questions in no time. Keep practicing, and you'll become a master of the distributive property in no time. Keep learning, keep practicing, and keep having fun with math! Your hard work and dedication will pay off, and you'll be well on your way to math mastery! You got this!