Solving -7 × 2 × (-9) A Step-by-Step Guide

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Hey guys! Today, we're going to dive into a super important math concept: multiplying integers. Specifically, we'll break down how to solve the problem -7 × 2 × (-9) step-by-step. Don't worry, it might look intimidating at first, but I promise it's totally manageable once you understand the basic rules. We'll go through each step slowly and clearly, so you can confidently tackle similar problems in the future. Whether you're a student struggling with math homework or just someone looking to brush up on your skills, this guide is for you. So, grab your pencils and let's get started!

Understanding the Basics of Integer Multiplication

Before we jump into the problem itself, let's quickly review the fundamental rules of integer multiplication. This is crucial because understanding these rules is the key to getting the correct answer. Remember, integers are whole numbers (no fractions or decimals) and can be positive, negative, or zero. The rules for multiplying integers revolve around how positive and negative signs interact.

Here's the core concept: When you multiply two numbers with the same sign (either both positive or both negative), the result is always positive. Think of it as the signs canceling each other out in a way. On the flip side, when you multiply two numbers with different signs (one positive and one negative), the result is always negative. It's like the negative sign 'wins' in this scenario. Understanding this positive/negative interaction is super important for solving any integer multiplication problem, including our example of -7 × 2 × (-9).

Let's break this down further with some simple examples. If we multiply 3 × 4, both numbers are positive, so the answer is a positive 12. Easy peasy! Now, if we multiply -3 × -4, both numbers are negative, but remember the rule: same signs, positive result. So, the answer is also a positive 12. See? The signs canceled out. But, if we multiply -3 × 4, we have different signs, so the result is a negative 12. Similarly, 3 × -4 also gives us a negative 12. These simple examples illustrate the basic principles that we'll apply to our main problem. We need to keep these rules at the forefront of our minds as we move through the steps, ensuring we handle the signs correctly at every stage. This not only helps in arriving at the correct numerical answer but also builds a solid understanding of the underlying mathematical logic. Keep practicing these basic scenarios, and you'll become a pro at integer multiplication in no time!

Step 1: Multiplying the First Two Integers (-7 and 2)

Okay, let's dive into the first step of solving -7 × 2 × (-9). We're going to start by focusing on the first two integers: -7 and 2. Remember, we're following the order of operations, which means we work from left to right when dealing with multiplication. So, the first calculation we need to do is -7 multiplied by 2. Now, think about the rules we just discussed. We have a negative number (-7) and a positive number (2). What happens when we multiply numbers with different signs? That's right, the result will be negative!

So, we know the answer will be negative, but what's the numerical value? We simply multiply the absolute values of the numbers, which are 7 and 2. 7 multiplied by 2 is 14. Now, we combine this numerical value with the negative sign we determined earlier. Therefore, -7 multiplied by 2 equals -14. It's crucial to keep track of the negative sign here. A common mistake people make is forgetting the sign, which can lead to the wrong answer. So, make it a habit to always check the signs before you do the multiplication and then apply the correct sign to your result. This simple step can save you a lot of headaches and ensure accuracy.

We've now successfully completed the first part of our problem. We've taken two integers, multiplied them, and correctly determined the sign of the result. This might seem like a small step, but it's a building block for the rest of the solution. We now have a new, simplified problem to tackle: -14 × (-9). This demonstrates the beauty of breaking down complex problems into smaller, manageable steps. By focusing on one multiplication at a time, we avoid getting overwhelmed and increase our chances of getting the correct answer. So, with this first step under our belts, we're well on our way to solving the entire problem. Let's move on to the next step and see how we handle multiplying -14 by -9.

Step 2: Multiplying the Result by the Third Integer (-14 and -9)

Alright, we're on to the second step, and it's time to multiply our previous result, -14, by the third integer, which is -9. So, we're looking at -14 × (-9). Again, the sign rules are going to be super important here. Take a moment to think about what happens when we multiply two negative numbers. Remember our rule: when we multiply two numbers with the same sign (in this case, both negative), the result is always positive. This is great news because we know our final answer will be positive, which simplifies things a bit!

Now, let's focus on the numerical value. We need to multiply 14 by 9. If you're comfortable with your multiplication facts, you might already know that 14 × 9 equals 126. If not, no worries! You can always break it down further. For example, you could think of 14 × 9 as (10 × 9) + (4 × 9). 10 × 9 is 90, and 4 × 9 is 36. Add those together, and you get 90 + 36 = 126. There are different ways to approach multiplication, so find the method that works best for you. The key is to be accurate and methodical. Now that we know 14 multiplied by 9 is 126, we can confidently say that -14 multiplied by -9 is positive 126. We've handled the signs correctly, performed the multiplication, and arrived at our final answer!

This step really highlights the power of the sign rules in integer multiplication. By remembering that two negatives make a positive, we avoid a common pitfall and arrive at the correct answer. It's also a good reminder that there are often multiple ways to perform multiplication. Whether you use mental math, break down the numbers, or use a calculator, the important thing is to understand the underlying process and be confident in your result. So, we've successfully multiplied -14 by -9 and gotten a positive 126. That means we've solved the entire problem! Let's recap our steps and celebrate our accomplishment.

Final Answer: 126

Woohoo! We've made it to the end, and we've successfully solved the problem -7 × 2 × (-9). Let's quickly recap the steps we took to get there, just to solidify our understanding. First, we multiplied -7 by 2, which gave us -14. Remember, a negative times a positive results in a negative. Then, we took that result, -14, and multiplied it by -9. This is where our rule of a negative times a negative equaling a positive came into play. We multiplied 14 by 9 to get 126, and since both numbers were negative, our final answer is positive 126.

So, the final answer to -7 × 2 × (-9) is 126. Great job if you followed along and understood each step! Integer multiplication might seem tricky at first, but with practice and a good grasp of the sign rules, you can conquer any problem that comes your way. The key is to break it down into smaller, manageable steps, focus on the signs, and double-check your work. Don't be afraid to use different multiplication methods to find what works best for you, whether it's mental math, breaking down the numbers, or using a calculator for more complex calculations. The important thing is to understand the underlying concept and be able to apply it confidently.

This problem also illustrates the associative property of multiplication, which means that the way we group the numbers doesn't change the final answer. We could have multiplied 2 × (-9) first and then multiplied the result by -7, and we would still have gotten 126. This flexibility can be helpful when tackling more complex problems. Keep practicing, and you'll become an integer multiplication master in no time! You've got this!

Tips and Tricks for Integer Multiplication

Now that we've solved our example problem, let's talk about some general tips and tricks that can help you master integer multiplication. These tips are designed to make the process smoother, more efficient, and less prone to errors. Whether you're facing simple multiplications or more complex problems with multiple integers, these strategies will come in handy.

  • Master the Sign Rules: I can't stress this enough – knowing the sign rules inside and out is the most crucial aspect of integer multiplication. Remember, same signs (positive × positive or negative × negative) result in a positive answer, while different signs (positive × negative or negative × positive) result in a negative answer. Make flashcards, practice with examples, or create a mental checklist – whatever works best for you to internalize these rules. When you approach a problem, the very first thing you should do is determine the sign of the final answer. This will give you a frame of reference and help you avoid careless mistakes.

  • Break Down Large Numbers: Multiplying larger numbers can be intimidating, but you can make it easier by breaking them down. For example, if you need to multiply 15 by -8, you could think of it as (10 × -8) + (5 × -8). This breaks the problem into two smaller, more manageable multiplications. Then, you can easily calculate -80 + (-40), which equals -120. This technique is especially helpful for mental math or when you don't have a calculator handy. It allows you to focus on smaller chunks of the problem and reduces the chance of errors.

  • Use the Commutative and Associative Properties: The commutative property of multiplication states that the order of the numbers doesn't matter (a × b = b × a). The associative property states that the grouping of the numbers doesn't matter ((a × b) × c = a × (b × c)). You can use these properties to rearrange the problem and make it easier to solve. For instance, in our original problem -7 × 2 × (-9), we multiplied -7 by 2 first. But we could have also multiplied 2 by -9 first and then multiplied the result by -7. The final answer will be the same. Look for groupings that make the multiplication simpler.

  • Practice Regularly: Like any math skill, mastering integer multiplication requires practice. The more you practice, the more comfortable you'll become with the rules and techniques. Start with simple problems and gradually work your way up to more complex ones. There are tons of online resources, worksheets, and textbooks that offer practice problems. You can also create your own problems and challenge yourself. Consistent practice is the key to building confidence and fluency in integer multiplication.

  • Double-Check Your Work: Always take a few moments to double-check your work, especially when dealing with negative signs. It's easy to make a small mistake, and a quick review can catch those errors before they become a problem. Check the signs, make sure you've multiplied correctly, and verify your final answer. If possible, try solving the problem using a different method or breaking it down in a different way to confirm your result. This extra step can save you from making mistakes and help you build a stronger understanding of the process.

By incorporating these tips and tricks into your study routine, you'll not only improve your integer multiplication skills but also develop a more confident and strategic approach to problem-solving in mathematics. Remember, it's all about understanding the fundamentals, practicing consistently, and taking your time to ensure accuracy. You've got this!

Real-World Applications of Integer Multiplication

Okay, so we've learned the rules and tips for integer multiplication, but you might be wondering, “When am I ever going to use this in real life?” That's a valid question! While it might not seem immediately obvious, integer multiplication actually pops up in various real-world scenarios. Understanding these applications can make the concept more relatable and help you appreciate its importance. Let's explore a few examples.

  • Finance and Budgeting: Integers are commonly used in finance to represent gains and losses. For instance, if you deposit $50 into your bank account, that's a positive integer (+50). If you withdraw $30, that's a negative integer (-30). Now, imagine you're consistently saving $25 each week for 8 weeks. This is a straightforward positive integer multiplication: 25 × 8. But, let's say you have a recurring expense of $15 per week (like a streaming subscription) for 12 weeks. This is where negative integers come in. The total expense can be calculated as -15 × 12, which gives you -180. So, integer multiplication helps you calculate total savings, expenses, and overall financial status. It's crucial for budgeting, tracking investments, and making informed financial decisions.

  • Temperature Changes: Temperature scales often dip below zero, especially in colder climates. These sub-zero temperatures are represented by negative integers. If the temperature is dropping at a rate of 3 degrees per hour (-3) for 5 hours, you can use integer multiplication to find the total temperature change. -3 × 5 equals -15, meaning the temperature has dropped by 15 degrees. This concept is not only useful for weather forecasting but also in various scientific and engineering applications where temperature control is critical.

  • Depth and Altitude: In fields like oceanography and aviation, integers are used to represent depth below sea level (negative) and altitude above sea level (positive). Imagine a submarine diving at a rate of 10 meters per minute (-10) for 20 minutes. The total depth reached can be calculated as -10 × 20, which is -200 meters. Similarly, an airplane climbing at a rate of 500 feet per minute (+500) for 30 minutes would have gained 500 × 30 = 15,000 feet in altitude. Integer multiplication helps in determining positions, distances, and changes in elevation in these contexts.

  • Game Development and Programming: Integer multiplication is fundamental in computer programming and game development. For example, if you're programming the movement of an object in a game, you might use integers to represent its position, velocity, and direction. If an object is moving at a speed of -5 units per second (negative indicating direction) for 10 seconds, the total displacement can be calculated as -5 × 10 = -50 units. This concept is essential for creating realistic movements, animations, and interactions within the game environment.

  • Scoring Systems: Many games and sports use scoring systems that involve both positive and negative points. Think about a trivia game where you get +10 points for a correct answer and -5 points for a wrong answer. If you answer 8 questions correctly and 3 questions incorrectly, you can use integer multiplication to calculate your total score. (8 × 10) + (3 × -5) = 80 + (-15) = 65. Integer multiplication allows for accurate tracking of scores and performance in various competitive settings.

These are just a few examples, but they illustrate that integer multiplication is not just an abstract mathematical concept. It's a practical tool that helps us understand and solve problems in a wide range of real-world situations. By recognizing these applications, you can appreciate the relevance of learning integer multiplication and its potential to help you in your daily life and future endeavors.