Solving $4((-3+6)(-5+8)+3(-2+5)) \div(-6)$ Step-by-Step

Hey guys! Ever stumbled upon a mathematical expression that looks like it belongs in a sci-fi movie rather than a textbook? Well, fear not! Today, we're going to break down one of those seemingly complex expressions into bite-sized, easily digestible pieces. We're tackling the expression $4((-3+6)(-5+8)+3(-2+5)) \div(-6)$. Sounds intimidating, right? But trust me, with a methodical approach and a sprinkle of mathematical magic, we'll conquer this beast. Think of it as a puzzle, and each step we take is a piece falling into place. We’ll be using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to guide us through this mathematical journey. So, buckle up, grab your calculators (or your mental math muscles), and let's dive in!

Understanding the Order of Operations (PEMDAS)

Before we even think about touching the expression, let's quickly recap the golden rule of math – the order of operations, or PEMDAS. This nifty acronym is our roadmap, ensuring we solve the expression in the correct sequence. P stands for Parentheses, meaning we tackle anything inside parentheses first. Then comes E for Exponents, which we'll deal with if there are any powers or roots. M and D represent Multiplication and Division, which we perform from left to right. Finally, A and S stand for Addition and Subtraction, also done from left to right. Mastering PEMDAS is like having a superpower in the math world; it’s the key to unlocking the solution to any complex expression. This order prevents ambiguity and ensures everyone arrives at the same answer, regardless of who's solving the problem. Imagine if we didn't have this order – we'd be in mathematical chaos! So, let’s keep PEMDAS in our back pocket as we navigate this expression, making sure we don't miss a step. Think of it as the secret sauce that makes our mathematical dish come out perfectly every time. And remember, practice makes perfect! The more you use PEMDAS, the more it becomes second nature, and the easier these expressions will seem.

Step 1: Simplifying Inside the Parentheses

Okay, let's get our hands dirty! Following PEMDAS, the first thing we need to address is the parentheses. We've got a couple of sets of parentheses within our main expression: $(-3+6)$, $(-5+8)$, and $(-2+5)$. Let's simplify each of these individually. For $(-3+6)$, we're essentially adding a negative number to a positive one. Think of it like having 6 dollars and spending 3; you're left with 3. So, $(-3+6) = 3$. Next up, $(-5+8)$. Again, we're adding a negative and a positive. This time, imagine having 8 dollars and spending 5; you'll have 3 dollars remaining. Thus, $(-5+8) = 3$. Lastly, we have $(-2+5)$. This is similar to the previous examples; 5 dollars minus 2 dollars leaves us with 3 dollars. Therefore, $(-2+5) = 3$. Now, our expression looks a little less intimidating: $4((3)(3)+3(3)) \div(-6)$. See how we've replaced those initial parentheses with their simplified values? We're making progress, guys! Each small step we take brings us closer to the final answer. Remember, breaking down a complex problem into smaller, manageable chunks is a fantastic strategy, not just in math, but in life too! So, pat yourselves on the back – we've conquered the first hurdle. What was once a jumble of numbers and symbols is now a clearer, more approachable expression. Let's keep this momentum going!

Step 2: Tackling the Inner Expression

Great job simplifying the initial parentheses! Now, let's zoom in on what's happening inside the remaining parentheses: $((3)(3)+3(3))$. According to PEMDAS, we need to handle multiplication before we can even think about addition. So, let's tackle those multiplications first. We have $(3)(3)$, which simply means 3 multiplied by 3, giving us 9. Then we have $3(3)$, which is also 3 multiplied by 3, resulting in 9. Now our expression within the parentheses looks like this: $(9+9)$. Much simpler, right? We've transformed multiplication into addition, making our lives easier. Next up, we perform the addition. $9+9$ is a classic sum, and it equals 18. So, we've successfully simplified the entire expression within the parentheses to just 18. Our full expression now reads: $4(18) \div(-6)$. Look how far we've come! We've chipped away at the complexity, one step at a time. This is a testament to the power of methodical problem-solving. By following PEMDAS and breaking down the expression, we've made it significantly more manageable. It's like decluttering your room – once you organize everything, it's much easier to find what you need. And in this case, what we need is the final answer! So, let's carry on with this momentum and move on to the next step.

Step 3: Multiplication and Division

Alright, we're on the home stretch now! Our expression stands at $4(18) \div(-6)$. PEMDAS tells us that we handle multiplication and division from left to right. So, first up is the multiplication: $4(18)$. This means 4 multiplied by 18. If you're a whiz with mental math, you might already know the answer. If not, no worries! You can break it down: 4 times 10 is 40, and 4 times 8 is 32. Add those together, and you get 72. So, $4(18) = 72$. Our expression now looks like this: $72 \div(-6)$. We've transformed a multiplication problem into a division problem. Now comes the final act: division. We need to divide 72 by -6. Remember, when you divide a positive number by a negative number, the result is negative. So, we know our answer will be negative. Now, how many times does 6 go into 72? If you know your multiplication tables, you'll recall that 6 times 12 is 72. Therefore, $72 \div(-6) = -12$. And there we have it! The final answer to our complex expression is -12. We've conquered the challenge! Give yourselves a round of applause, guys. We took a seemingly daunting expression and, by following the rules of PEMDAS and breaking it down step by step, arrived at a clear, concise solution.

Final Answer: -12

So, to recap, we started with the expression $4((-3+6)(-5+8)+3(-2+5)) \div(-6)$ and, by systematically applying the order of operations (PEMDAS), we've successfully navigated through each step. We first simplified the expressions within the parentheses, then tackled the inner expression involving multiplication and addition, and finally performed the remaining multiplication and division. This journey demonstrates the power of methodical problem-solving. By breaking down a complex problem into smaller, manageable steps, we can conquer even the most intimidating mathematical challenges. The final answer, after all our hard work, is -12. Isn't it satisfying to see how a jumble of numbers and symbols can be tamed and reduced to a single, elegant solution? Remember, math isn't about magic; it's about understanding the rules and applying them consistently. And with practice and a bit of perseverance, you can become a math master too! So, the next time you encounter a daunting expression, don't shy away. Embrace the challenge, remember PEMDAS, and break it down. You've got this!