Multiplying Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of multiplying expressions, specifically tackling a problem like (9.9x)(9x). It might seem a little daunting at first, but trust me, it's a piece of cake once you break it down. We're going to explore this step by step, making sure you grasp the fundamentals and can confidently solve similar problems. So, let's get started and unravel this together! This will not only clarify how to multiply the given expression but also provide a solid foundation for more complex algebraic manipulations. We'll be using the commutative and associative properties of multiplication, which are basically fancy terms for saying we can rearrange and group numbers as we like. And don't worry, I will make sure we do this step by step.

First, let's talk about the basics of multiplying variables. When we see a term like 'x', it really means '1x'. Also, when we multiply variables with exponents, we add the exponents. For instance, x * x = x². It's a fundamental concept, and once you get the hang of it, you'll be well on your way to mastering algebra. Now, let’s get into the specifics of multiplying the given expression. When multiplying expressions, like (9.9x)(9x), we need to multiply the coefficients (the numbers in front of the variables) and then multiply the variables together. If there are exponents, we need to know how to add and multiply them properly. Remember, multiplying is commutative and associative, so we can change the order and group terms as we like, which greatly simplifies the problem. In this case, we're dealing with a single variable, x, but the principles remain the same for more complex expressions with multiple variables or constants. This is an essential skill, and the steps we'll use here are adaptable to various algebraic problems.

Now, let's tackle the given problem of multiplying the expressions (9.9x)(9x). This is where the magic happens! We'll go step by step, ensuring you understand each move and its rationale. The key is to separate the numerical coefficients from the variables, multiply them, and then combine the variables. The goal is to simplify it as much as possible. This approach not only solves the problem but also serves as a template for other similar algebraic exercises. The power of this skill extends far beyond simple multiplication; it is crucial for solving equations, simplifying expressions, and understanding complex mathematical concepts. The process we will be discussing is very useful. Let's delve into the actual multiplication, shall we?

Step-by-Step Guide to Multiplying $(9.9x)(9x)$

Alright, guys, let's break down the multiplication of (9.9x)(9x) into easy-to-follow steps. We will go slowly, so everyone gets it, I promise! No need to feel intimidated; we'll navigate through each part carefully, ensuring you grasp the core principles.

Step 1: Multiply the Coefficients

The first thing we need to do is multiply the numerical coefficients. In our expression (9.9x)(9x), the coefficients are 9.9 and 9. Multiply these two numbers together. You can either use a calculator or do it manually. I recommend practicing by hand as it helps you become more familiar with the numbers, but a calculator is always handy. When you multiply 9.9 by 9, you get 89.1. Keep this number handy because we'll be using it very soon. You can see this process as the first stage of the simplification, we're taking the complicated part and making it easier to manage. This is a very essential step. So, in our problem, this step looks like: 9.9 * 9 = 89.1.

Step 2: Multiply the Variables

Now it's time to multiply the variables together. Here, we only have 'x' in both terms. So, we multiply x by x. Remember the basic rule of exponents? When you multiply like variables, you add their exponents. Since 'x' is the same as x¹, then x * x = x¹ * x¹ = x². If we had different variables like x and y, we'd simply write them side by side, such as xy. Combining variables correctly is essential for getting the right answer and is also critical for solving more complex equations down the road. This step might seem simple here, but it's a vital part of the algebraic manipulation.

Step 3: Combine the Results

Finally, we combine the results from Steps 1 and 2. We've calculated the product of the coefficients (89.1) and the variables (x²). Now, we just put them together. The final answer is 89.1x². You got it! We have successfully multiplied (9.9x)(9x)! The ability to combine coefficients and variables is fundamental to algebra and will be used in a lot of more complex problems later on. Always double-check your work to catch any small mistakes. And congratulations, you've just conquered this problem!

Understanding the Properties Used

During this multiplication process, we used several important mathematical properties that make it all work seamlessly. These properties are the backbone of algebra and are used constantly in more complex problems. Understanding them is key.

The Commutative Property of Multiplication

The commutative property states that the order in which you multiply numbers doesn't affect the result. For example, a * b = b * a. In our case, this means we could have rewritten the problem as (9x)(9.9x) without changing the final answer. This property gives us the flexibility to rearrange terms to make the multiplication easier.

The Associative Property of Multiplication

The associative property states that the way you group numbers in a multiplication problem doesn't change the answer. For example, (a * b) * c = a * (b * c). This means that when multiplying several numbers, we can group them as we wish without impacting the outcome. This is helpful when you are simplifying the problem, especially when using larger numbers.

Distributive Property (Not Directly Used, But Important)

Although not directly used in the problem, the distributive property is very important in algebra. The distributive property says that a(b + c) = ab + ac. This property is very useful when dealing with expressions involving parentheses, allowing you to multiply a term by each term inside the parentheses. So keep in mind, even if we did not use it in this specific problem, it is very essential in algebra.

Practice Problems and Tips

Now that we've walked through this problem, let's provide you with some practice problems and tips to help you master this skill. Practice makes perfect, and the more you practice, the more comfortable and confident you'll become in solving similar problems. I will also give you some tips on how to improve your skills.

Practice Problems

Here are some practice problems for you to try out. I suggest you solve each problem independently before checking the answers. This will really help you to get a better understanding of the concepts.

1. (3x)(5x)
2. (2.5x)(4x)
3. (7x)(-2x)
4. (-4.5x)(-3x)
5. (1/2 x)(6x)

Tips for Success

Here are some tips to help you succeed in algebra and similar problems. These will help you to understand and get better results in math problems:

• Always write down each step: This helps you keep track of your work and reduces the chance of making mistakes. It also helps in identifying where you might have gone wrong.
• Double-check your answers: Always double-check your work, particularly when dealing with negative signs or decimal numbers. Little mistakes can lead to major errors.
• Practice regularly: Consistent practice is key to mastering algebra. Try solving a few problems every day to strengthen your skills.
• Understand the basics: Ensure you understand the basic concepts of exponents and variables. These are the building blocks of algebra.
• Use online resources: Use online calculators and resources to check your work, but try to solve the problems manually first to truly understand the process.

Conclusion

Well, guys, there you have it! We've successfully navigated through the multiplication of expressions, specifically tackling the problem of (9.9x)(9x). We broke it down into simple steps, explored the underlying mathematical properties, and offered you some practice problems and tips to solidify your understanding. Remember, the key to mastering algebra is understanding the basic concepts and consistent practice. Keep at it, and you'll find yourself acing these types of problems in no time. If you have any questions, feel free to ask! Happy calculating!