Solve 12-{-2+3.[4-(-8+12)+]-2}+3: A Step-by-Step Guide

Hey guys! Today, we're diving headfirst into a fascinating mathematical expression: 12-{-2+3.[4-(-8+12)+]-2}+3. This might look like a jumble of numbers and symbols at first glance, but trust me, it's a puzzle just waiting to be solved. We're going to break it down step by step, so you'll not only understand the solution but also the logic behind it. Let's get started and make math a little less intimidating and a lot more fun!

Understanding the Order of Operations

Before we even think about plugging in numbers, it's crucial to understand the order of operations. Think of it as the golden rule of math – a set of instructions that tells us which calculations to perform first. The most common mnemonic for remembering this is PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Why is this so important? Imagine trying to follow a recipe without knowing the order of the steps. You might end up with a culinary disaster! Similarly, in math, if we don't follow the order of operations, we're likely to arrive at the wrong answer. For our expression, this means we'll tackle the parentheses first, then any multiplication, and finally, addition and subtraction. This structured approach ensures we solve the equation accurately and efficiently.

Parentheses First: The Innermost Layer

Okay, let's zoom in on the heart of our expression: 12-{-2+3.[4-(-8+12)+]-2}+3. The first thing that jumps out is the nested parentheses. We've got parentheses inside brackets, which are inside curly braces – it's like a mathematical Russian doll! According to PEMDAS, we need to start with the innermost set of parentheses first. In our case, that's (-8+12). Think of this as adding -8 and 12. If you're 8 steps behind on a number line and then move 12 steps forward, where do you end up? You land on 4! So, (-8+12) = 4. Now, we can replace that part of the expression with 4, making it a little less complex. This might seem like a small step, but it's a crucial one in simplifying the overall problem. By focusing on the innermost operations, we break down the problem into manageable chunks, making it far less overwhelming.

Moving Outwards: Tackling Brackets

Great job on conquering the first set of parentheses! Now, our expression looks like this: 12-{-2+3.[4-(4)+]-2}+3. See how things are already getting simpler? Next up, we need to deal with the brackets [4-(4)+]. Inside the brackets, we have a subtraction operation: 4 - 4. This is pretty straightforward – any number subtracted from itself equals zero. So, 4 - 4 = 0. Now, we can replace the brackets with 0, further simplifying our expression. This step-by-step approach is key to mastering complex mathematical problems. We're not trying to do everything at once; instead, we're methodically working our way through each operation, making sure we don't miss a thing. Remember, patience and precision are your best friends in math!

Curly Braces: The Next Frontier

Alright, we're making fantastic progress! Our expression is now shaping up nicely: 12-{-2+3.[0]-2}+3. The brackets are gone, and we're ready to tackle the curly braces {-2+3.[0]-2}. Remember, according to PEMDAS, multiplication comes before addition and subtraction. So, within the curly braces, we need to handle 3.[0] first. Any number multiplied by zero is zero, so 3 * 0 = 0. Now, our expression inside the curly braces simplifies to -2 + 0 - 2. Let's tackle this from left to right. -2 + 0 is simply -2. Then, -2 - 2 equals -4. So, the entire expression within the curly braces simplifies to -4. This is a significant step forward. We've managed to condense a complex set of operations into a single number. By focusing on the order of operations and taking our time, we're making this problem much easier to solve.

Multiplication and Addition/Subtraction: The Final Showdown

With the curly braces resolved, our expression is looking much more manageable: 12 - {-4} + 3. Now, we're left with subtraction and addition. But wait, what about that -{-4}? Remember that subtracting a negative number is the same as adding its positive counterpart. So, -{-4} becomes +4. Our expression now reads: 12 + 4 + 3. This is a simple addition problem! Let's add the numbers from left to right. 12 + 4 equals 16. Then, 16 + 3 equals 19. So, the final result of our complex expression is 19. Congratulations, we've made it! We've successfully navigated through the parentheses, brackets, curly braces, multiplication, and addition/subtraction to arrive at the solution. This is a testament to the power of understanding the order of operations and breaking down problems into smaller, more manageable steps.

The Final Result and What We Learned

So, after all that mathematical maneuvering, we've arrived at our final answer: 12-{-2+3.[4-(-8+12)+]-2}+3 = 19. Woohoo! Give yourself a pat on the back. You've not only solved a complex mathematical expression but also reinforced some crucial mathematical principles. The biggest takeaway here is the importance of the order of operations (PEMDAS). Without it, we'd be lost in a sea of numbers and symbols. Remember, parentheses (and their nested cousins) always come first, followed by exponents, then multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). This order is the key to unlocking any mathematical puzzle.

But it's not just about memorizing PEMDAS. It's about understanding why it works. By prioritizing operations within parentheses, we're essentially isolating parts of the expression and simplifying them before they interact with the rest of the equation. This prevents us from making mistakes and ensures we're always working with the most simplified version of the problem. Similarly, performing multiplication and division before addition and subtraction ensures that we're properly scaling and distributing values before we combine them. Math is not just about getting the right answer; it's about understanding the process and the logic behind it. And that's what we've accomplished today. We've not only solved a problem but also deepened our understanding of how math works. Keep practicing, keep exploring, and most importantly, keep having fun with math!