Simplifying (m + 4m)(5 + 4): A Step-by-Step Guide

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Hey guys! Ever stumbled upon an algebraic expression that looks a bit intimidating? Don't worry; we've all been there. Today, we're going to break down a classic example: (m + 4m)(5 + 4). This might seem complex at first glance, but with a few simple steps, we'll have it simplified in no time. So, grab your pencils and let's dive into the world of algebraic simplification!

Understanding the Basics of Algebraic Expressions

Before we jump into the problem, it's crucial to understand what we're dealing with. An algebraic expression is a combination of variables (like 'm'), constants (like 4 and 5), and mathematical operations (like addition and multiplication). Simplifying an expression means rewriting it in its most basic form, while still maintaining its original value. This often involves combining like terms and performing operations in the correct order. Remember the good old PEMDAS/BODMAS rule? Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is our roadmap for simplifying expressions!

When dealing with algebraic expressions, especially those involving variables, the distributive property often comes into play. This property states that a(b + c) = ab + ac. In simpler terms, it means that you can multiply a single term by each term inside a set of parentheses. Mastering this property is essential for simplifying more complex expressions, and it's something we'll be using implicitly in our example today. Moreover, understanding the concept of like terms is crucial. Like terms are terms that have the same variable raised to the same power. For example, 3x and 5x are like terms because they both have the variable 'x' raised to the power of 1. We can combine like terms by simply adding or subtracting their coefficients (the numbers in front of the variables). This foundational knowledge will make simplifying expressions like ours much easier and more intuitive.

Step 1: Simplify within the Parentheses

The first step in simplifying any expression, as dictated by PEMDAS/BODMAS, is to tackle the parentheses. We have two sets of parentheses in our expression: (m + 4m) and (5 + 4). Let's start with the first one.

Inside the first parenthesis, we have "m + 4m." Notice that both terms contain the variable "m." This means they are like terms, and we can combine them. Think of "m" as "1m." So, we have 1m + 4m. Adding the coefficients (the numbers in front of the 'm'), we get 1 + 4 = 5. Therefore, m + 4m simplifies to 5m.

Now, let's move on to the second set of parentheses: (5 + 4). This is a straightforward addition problem. 5 + 4 equals 9. So, (5 + 4) simplifies to 9. By simplifying within the parentheses first, we've reduced the complexity of the original expression, making it much easier to work with in the subsequent steps. This initial simplification is a crucial step in solving algebraic expressions, and it highlights the importance of following the correct order of operations. It sets the stage for the next steps, where we'll further simplify the expression by performing multiplication.

Step 2: Multiply the Simplified Terms

After simplifying the expressions within the parentheses, our original expression (m + 4m)(5 + 4) has now been reduced to (5m)(9). This is much more manageable, right? Now, we simply need to multiply these two terms together. Remember, when we have a term like "5m," it means 5 multiplied by m. So, we're essentially multiplying 5m by 9.

To do this, we multiply the coefficients (the numbers in front of the variables). In this case, we multiply 5 by 9. 5 multiplied by 9 equals 45. So, the result of multiplying 5m by 9 is 45m. This step showcases how simplifying within parentheses first can lead to a much clearer and easier multiplication problem. Instead of dealing with a more complex expression, we've boiled it down to a simple multiplication of a coefficient and a variable. Understanding this process is key to tackling more challenging algebraic simplifications in the future. This multiplication is the final step in simplifying the expression, giving us our final answer.

Step 3: The Final Simplified Expression

We've reached the final step! After simplifying within the parentheses and then multiplying the resulting terms, we have arrived at our simplified expression. From the initial expression (m + 4m)(5 + 4), we first simplified the parentheses to get (5m)(9). Then, we multiplied 5m by 9. The result of this multiplication is 45m.

Therefore, the simplified form of the expression (m + 4m)(5 + 4) is 45m. This is our final answer! We've successfully taken a seemingly complex expression and reduced it to its simplest form. By following the order of operations and combining like terms, we were able to efficiently simplify the expression. This result, 45m, represents the most concise and understandable form of the original expression, making it easier to work with in further mathematical operations or problem-solving scenarios. This final simplification demonstrates the power of algebraic manipulation and the importance of understanding fundamental mathematical principles.

Key Takeaways for Simplifying Expressions

Simplifying algebraic expressions might seem daunting at first, but with a clear understanding of the rules and a step-by-step approach, it becomes a manageable process. Here are some key takeaways to keep in mind when tackling similar problems:

  1. Follow the Order of Operations: Always remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is the golden rule for simplifying expressions.
  2. Combine Like Terms: Identify terms that have the same variable raised to the same power. You can add or subtract their coefficients to combine them into a single term. This significantly reduces the complexity of the expression.
  3. Master the Distributive Property: The distributive property (a(b + c) = ab + ac) is a powerful tool for eliminating parentheses and simplifying expressions. Practice using it to expand expressions and combine terms.
  4. Break It Down: Complex expressions can be intimidating. Break the problem down into smaller, more manageable steps. Simplify within parentheses first, then perform multiplication and division, and finally, addition and subtraction.
  5. Practice, Practice, Practice: The more you practice simplifying expressions, the more comfortable and confident you'll become. Start with simple problems and gradually work your way up to more challenging ones.

By keeping these takeaways in mind, you'll be well-equipped to tackle a wide range of algebraic expressions and simplify them with ease. Remember, algebra is like a puzzle, and simplification is the key to unlocking the solution!

Let's Practice! More Examples

To solidify your understanding of simplifying expressions, let's look at a couple more examples. Working through these will help you get a better grasp of the process and the key principles involved.

Example 1: Simplify 2(x + 3) - 4x

  • Step 1: Distribute: Apply the distributive property to eliminate the parentheses: 2 * x + 2 * 3 = 2x + 6. So the expression becomes 2x + 6 - 4x.
  • Step 2: Combine Like Terms: Identify and combine like terms. We have 2x and -4x, which are like terms. Combining them gives us 2x - 4x = -2x. The expression now looks like -2x + 6.
  • Final Answer: The simplified expression is -2x + 6.

Example 2: Simplify 3(2y - 1) + 5(y + 2)

  • Step 1: Distribute: Apply the distributive property to both sets of parentheses: 3 * 2y - 3 * 1 = 6y - 3 and 5 * y + 5 * 2 = 5y + 10. The expression becomes 6y - 3 + 5y + 10.
  • Step 2: Combine Like Terms: Combine like terms. We have 6y and 5y, which combine to 11y. We also have -3 and +10, which combine to 7. The expression now looks like 11y + 7.
  • Final Answer: The simplified expression is 11y + 7.

By working through these examples, you can see how the same principles we used for our original problem apply to other expressions. Remember to focus on the order of operations, combining like terms, and using the distributive property. With practice, you'll become a pro at simplifying expressions!

Conclusion: You've Got This!

Simplifying algebraic expressions is a fundamental skill in mathematics. We've tackled the expression (m + 4m)(5 + 4) together, breaking down each step and highlighting the key concepts involved. From understanding the order of operations (PEMDAS/BODMAS) to combining like terms and mastering the distributive property, you now have a solid foundation for simplifying a variety of expressions.

Remember, practice is key! The more you work with algebraic expressions, the more comfortable and confident you'll become. Don't be afraid to tackle challenging problems; break them down into smaller steps and apply the principles we've discussed. With patience and persistence, you'll master the art of simplification and unlock a deeper understanding of algebra. So go forth and simplify, guys! You've got this!