Simplify: 4z + (4z + Ab) + (3a9 - 4ab)

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Alright, guys, let's break down this math problem step by step. We've got the expression 4z + (4z + ab) + (3a9 - 4ab) and our mission, should we choose to accept it, is to simplify it. Don't worry, it's not as daunting as it looks! We'll walk through it together, making sure everyone understands each part. Remember, math is like building blocks; once you understand the basics, you can construct all sorts of amazing things. So, let’s get started and turn this equation into something super simple and easy to handle!

Step 1: Remove Parentheses

The first thing we need to do is get rid of those parentheses. When there's a plus sign in front of the parentheses, like in our case, we can just remove them without changing anything inside. So, 4z + (4z + ab) + (3a9 - 4ab) becomes 4z + 4z + ab + 3a9 - 4ab. Easy peasy, right? Removing parentheses is like decluttering your room; it makes everything much easier to see and organize. And in math, just like in life, organization is key! Now that we've cleared the way, let's move on to the next step and start combining like terms.

Step 2: Combine Like Terms

Now comes the fun part – combining like terms! This means we're going to group together all the terms that have the same variable. In our expression 4z + 4z + ab + 3a9 - 4ab, we have two terms with z and two terms with ab. Let's start with the z terms: 4z + 4z. When you add these together, you get 8z. Think of it like having four apples and then getting four more apples; now you have eight apples! Next, let's combine the ab terms: ab - 4ab. This is like having one 'ab' and taking away four 'ab's, which leaves you with -3ab. And finally, we have the term 3a9, which doesn't have any other like terms, so it stays as it is. So, after combining like terms, our expression looks like this: 8z - 3ab + 3a9. See how much simpler it's becoming? Combining like terms is like sorting your laundry; you put all the socks together, all the shirts together, and so on. It just makes everything neater and easier to manage.

Step 3: Rearrange the Terms (Optional)

This step is optional, but it can sometimes make the expression look cleaner. We can rearrange the terms so that the term with the highest power comes first. In our expression 8z - 3ab + 3a9, the term 3a9 might be considered to have a higher degree if 'a' is a variable, so we could write it as 3a9 + 8z - 3ab. However, the order doesn't really matter unless you're specifically asked to arrange it in a certain way. Think of it like arranging furniture in your room; you can put the sofa wherever you like, as long as it fits and looks good to you! So, whether you leave it as 8z - 3ab + 3a9 or rearrange it to 3a9 + 8z - 3ab, it's perfectly fine.

Final Answer

So, after simplifying the expression 4z + (4z + ab) + (3a9 - 4ab), we get 8z - 3ab + 3a9. And that's it! We've taken a somewhat complicated-looking expression and turned it into something much simpler. Remember, the key to simplifying expressions is to remove parentheses and combine like terms. With a little practice, you'll be simplifying expressions like a pro in no time! Great job, guys! You nailed it!

Key Concepts Recap

To make sure we're all on the same page, let's quickly recap the key concepts we used to solve this problem:

  • Removing Parentheses: When there's a plus sign in front of parentheses, you can simply remove them without changing the signs of the terms inside.
  • Combining Like Terms: This involves grouping together terms that have the same variable and exponent. For example, 4z and 4z are like terms because they both have the variable z raised to the power of 1.
  • Rearranging Terms: While not always necessary, rearranging terms can sometimes make an expression look cleaner and easier to understand. Generally, terms with higher powers are placed before terms with lower powers.

Common Mistakes to Avoid

Everyone makes mistakes, especially when they're learning something new. Here are some common mistakes to watch out for when simplifying expressions:

  • Forgetting to Distribute: If there's a negative sign in front of the parentheses, you need to distribute it to all the terms inside. For example, -(a + b) becomes -a - b.
  • Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can't combine 4z and 3ab because they have different variables.
  • Incorrectly Adding or Subtracting Coefficients: Make sure you're adding or subtracting the coefficients (the numbers in front of the variables) correctly. For example, 4z + 4z = 8z, not 16z^2.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

  1. Simplify: 5x + (2x - 3y) + (4y - x)
  2. Simplify: 3a - (2b + a) + (5b - 2a)
  3. Simplify: 6p + (3q - 2p) - (4q + p)

Work through these problems, and if you get stuck, review the steps we discussed earlier. Remember, practice makes perfect!

Real-World Applications

You might be wondering, "When will I ever use this in real life?" Well, simplifying expressions is actually a very useful skill in many different fields. For example:

  • Engineering: Engineers use simplified expressions to design structures, circuits, and machines.
  • Computer Science: Computer programmers use simplified expressions to write efficient code.
  • Finance: Financial analysts use simplified expressions to calculate investments and manage risk.
  • Everyday Life: Even in everyday life, you might use simplified expressions to calculate discounts, measure ingredients for a recipe, or plan a budget.

So, while it might not always be obvious, the ability to simplify expressions is a valuable skill that can help you in many different areas of life.

Conclusion

Great job, guys! You've successfully learned how to simplify the expression 4z + (4z + ab) + (3a9 - 4ab). Remember to remove parentheses, combine like terms, and watch out for common mistakes. With a little practice, you'll be simplifying expressions like a pro in no time. Keep up the great work, and don't be afraid to ask questions if you get stuck. Math can be challenging, but it's also incredibly rewarding when you finally understand something new. So, keep exploring, keep learning, and keep having fun with math!