Introduction
Hey guys! Today, we're diving into a fascinating mathematical problem that involves absolute values and integer operations. The expression we're tackling is |(-3+16)+(-12+5)|+(-(-6+5-5). Sounds intimidating? Don't worry, we'll break it down step by step in a friendly and easy-to-understand way. We'll explore how to simplify each part of the expression, handle the absolute values, and finally arrive at the solution. This isn't just about finding an answer; it's about understanding the process and the underlying mathematical principles. So, let's get started and unravel this mathematical puzzle together! We'll make sure to cover every detail, ensuring you not only get the answer but also grasp the method to solve similar problems in the future. Remember, mathematics is like a language, and once you understand the grammar, you can speak it fluently! We'll treat each component of the expression like a sentence, decoding it piece by piece until the whole meaning becomes clear. Whether you're a student brushing up on your algebra or just a curious mind, this explanation is tailored to help you understand and appreciate the beauty of mathematical expressions.
Breaking Down the Expression
Alright, let's get our hands dirty with the expression: |(-3+16)+(-12+5)|+(-(-6+5-5). Our mission is to simplify this step by step, making sure we don’t miss a thing. First up, we’ll handle the parentheses within the absolute value. Inside the first set of parentheses, we have -3 + 16. Think of this as starting at -3 on the number line and moving 16 steps to the right. What do we get? That’s right, 13. So, -3 + 16 = 13. Next, let's tackle the second set of parentheses: -12 + 5. Imagine starting at -12 and moving 5 steps to the right. We land at -7. So, -12 + 5 = -7. Now our expression looks a bit simpler: |13 + (-7)|+(-(-6+5-5). See? We're already making progress! This is how we approach complex problems – by breaking them down into smaller, manageable parts. It's like eating an elephant, one bite at a time! We're focusing on accuracy and clarity, ensuring each step is crystal clear before moving on. Remember, in math, the journey is just as important as the destination. Understanding each step helps build a solid foundation for more complex problems later on. This methodical approach not only helps in solving this particular problem but also cultivates a problem-solving mindset that’s valuable in all areas of life.
Simplifying Inside the Absolute Value
Okay, guys, let's keep the momentum going! We've simplified our expression to |13 + (-7)|+(-(-6+5-5). The next step is to deal with what’s inside the absolute value bars. We have 13 + (-7). Remember, adding a negative number is the same as subtracting its positive counterpart. So, this is the same as 13 - 7. Easy peasy, right? 13 - 7 equals 6. Now our expression looks even cleaner: |6|+(-(-6+5-5). We’re making great strides! The absolute value is starting to look less intimidating now that we've simplified what’s inside. This step highlights the importance of understanding basic arithmetic operations. Addition and subtraction are the building blocks upon which more complex mathematical concepts are built. By mastering these fundamentals, we can tackle seemingly difficult problems with confidence. And remember, practice makes perfect! The more you work with numbers and operations, the more intuitive they become. This part of the process also demonstrates the beauty of simplification in mathematics. By reducing the expression inside the absolute value to a single number, we’ve made the next step much clearer and easier to handle.
Dealing with the Second Part of the Expression
Alright, team, let's shift our focus to the second part of the expression: +(-(-6+5-5). We've handled the absolute value nicely, and now it's time to simplify this bit. First, let’s tackle the innermost parentheses: -6 + 5 - 5. We’ll go from left to right. -6 + 5 equals -1. So now we have -1 - 5. Think of this as starting at -1 on the number line and moving 5 steps further to the left. That puts us at -6. So, -6 + 5 - 5 = -6. Our expression now looks like this: |6|+(-(-6)). We're getting closer to the finish line! Notice how we're meticulously working through each operation, ensuring we don't make any careless mistakes. This careful approach is crucial in mathematics, where a small error can throw off the entire solution. We're also highlighting the importance of paying attention to signs – positive and negative. These little symbols can make a big difference in the outcome. By systematically simplifying the expression within the parentheses, we've transformed a potentially confusing set of operations into a single, manageable number. This skill of breaking down complex calculations is a cornerstone of mathematical proficiency.
Evaluating the Absolute Value and Final Simplification
Okay, let's bring it home! We've got our expression down to |6|+(-(-6)). The absolute value of a number is its distance from zero, and since 6 is 6 units away from zero, |6| = 6. Now we have 6 + (-(-6)). Next up, we need to deal with the -(-6). Remember, a negative times a negative is a positive. So, -(-6) is the same as +6. Our expression is now super simple: 6 + 6. And what does that equal? You guessed it – 12! So, the final answer to our problem is 12. Woohoo! We did it! We started with a seemingly complex expression and, by breaking it down step by step, we arrived at a clear and concise solution. This final step underscores the power of understanding fundamental mathematical principles. The concept of absolute value, the rules of multiplying negative numbers, and basic addition – these are the tools we used to unravel the mystery. It's also a testament to the importance of accuracy in calculations. Each step built upon the previous one, and a mistake at any point could have led to a wrong answer. This meticulous approach is what makes mathematics so precise and reliable.
Conclusion
So, guys, we've successfully evaluated the expression |(-3+16)+(-12+5)|+(-(-6+5-5) and found the answer to be 12! Give yourselves a pat on the back! We broke down a complex problem into manageable steps, simplified each part, and used our knowledge of absolute values and integer operations to reach the solution. Remember, the key to mastering math is to take things one step at a time, be meticulous in your calculations, and never be afraid to ask questions. Math can be challenging, but it's also incredibly rewarding when you finally crack a tough problem. Keep practicing, keep exploring, and keep that mathematical curiosity alive! We hope this detailed explanation has not only helped you understand this particular problem but has also given you a clearer understanding of how to approach similar problems in the future. Mathematics is a journey, not a destination, and every problem we solve is a step forward in our understanding of the world around us. Keep up the great work, and who knows what mathematical challenges we'll conquer next!