Solving 56 + 108 + 20 * 86 * 9 * 4: A Step-by-Step Guide

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Hey guys! Today, let's break down how to solve the mathematical expression 56 + 108 + 20 * 86 * 9 * 4. Don't worry, it might look intimidating at first, but we'll take it one step at a time. We'll use the order of operations, also known as PEMDAS/BODMAS, to make sure we get the correct answer. So, grab your calculators (or your mental math skills!) and let's dive in!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even touch the numbers, it's super important to understand the order of operations. This is the golden rule that tells us in what sequence we should perform mathematical operations. You might have heard of it as PEMDAS or BODMAS:

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

Think of it as a recipe – if you don't follow the steps in the right order, your final dish (or answer, in this case) won't be quite right. Ignoring this order can lead to a completely different result, and we definitely want to avoid that. So, let’s keep PEMDAS/BODMAS in mind as we tackle our expression.

Why is it so important, though? Imagine if we just went from left to right, doing operations as we see them. We'd end up with a completely wrong answer! The order of operations ensures that everyone solves the same expression in the same way, leading to a consistent and correct result. This is crucial in mathematics, science, and even programming, where precise calculations are essential. So, understanding this principle is a fundamental skill that will help you in many areas.

Let's illustrate with a simple example: 2 + 3 * 4. If we do addition first, we get 5 * 4 = 20. But if we do multiplication first (as PEMDAS/BODMAS tells us), we get 2 + 12 = 14. See the difference? The correct answer is 14, highlighting the importance of following the order of operations. This small example shows how critical it is to adhere to these rules, especially when dealing with more complex expressions.

Step 1: Identifying the Operations

Okay, let's take a good look at our expression: 56 + 108 + 20 * 86 * 9 * 4. What operations do we see here? We've got addition (+) and multiplication (*). According to PEMDAS/BODMAS, multiplication comes before addition, so we'll tackle those multiplications first. This is a crucial step because it sets the stage for the rest of the calculation. Misidentifying the order here would throw off the entire process, leading to an incorrect final answer.

It's like building a house – you can't put the roof on before the walls are up, right? Similarly, in math, we need to perform operations in the correct order to build towards the solution. In our expression, the multiplications are the foundational elements that we need to address before we can add the other numbers. This systematic approach not only ensures accuracy but also helps in breaking down complex problems into manageable parts. By focusing on one type of operation at a time, we reduce the chances of making errors and make the entire process much smoother.

Think of each operation as a different task. We prioritize the most important tasks first, which in this case are the multiplications. This way, when we get to the additions, we'll have the correct values to work with. It's all about strategic problem-solving, and identifying the operations is the first strategic move we make.

Step 2: Performing the Multiplications

Now for the fun part – the multiplications! We have 20 * 86 * 9 * 4. Since multiplication is associative (meaning we can group the numbers in any order), we can just go from left to right. Let's start with the first pair: 20 * 86. You can use a calculator for this, or if you're feeling brave, you can do it manually.

20 * 86 = 1720

Great! Now we have 1720 * 9 * 4. Let’s keep going. Next up is 1720 * 9:

1720 * 9 = 15480

We're almost there! Now we have 15480 * 4. One final multiplication:

15480 * 4 = 61920

Wow, that's a big number! But we've handled it like pros. We've successfully multiplied all the numbers together, and now we have the result of the multiplication part of our expression. This is a significant step, as we've simplified a large chunk of the original problem. By breaking it down into smaller multiplications, we made it much easier to manage and reduce the likelihood of errors. It's like cutting a large pizza into slices – each slice is easier to handle than the whole pizza!

Each of these individual multiplications is like a building block. We carefully calculate each one and then use the result to move on to the next step. This methodical approach is key to solving complex mathematical problems. It's not just about getting the final answer; it's also about understanding the process and building confidence in your ability to tackle even the most daunting-looking equations.

Step 3: Performing the Additions

Okay, we've conquered the multiplications, and now we're left with the additions. Our expression has been simplified to 56 + 108 + 61920. Just like with multiplication, addition is also associative, so we can go from left to right. Let's start with the first pair: 56 + 108.

56 + 108 = 164

Now we have 164 + 61920. Time for the final addition:

164 + 61920 = 62084

And there we have it! The result of the additions is 62084. We've successfully combined the numbers that were initially separated by addition signs. This step is like putting the finishing touches on a painting – it brings all the elements together to create the final piece. After carefully performing the multiplications, this final addition step is relatively straightforward, but it's still crucial to ensure accuracy.

Just as we broke down the multiplications, we're handling the additions in a similar manner. By adding the numbers in pairs, we make the process more manageable and less prone to errors. It's like climbing a staircase – we take it one step at a time, making sure each step is secure before moving on to the next. This methodical approach is a valuable skill in mathematics and in life in general, as it allows us to tackle complex challenges by breaking them down into simpler tasks.

Step 4: The Final Answer

Drumroll, please! We've reached the end of our mathematical journey. After carefully following the order of operations and performing all the necessary calculations, we've arrived at our final answer. So, what is 56 + 108 + 20 * 86 * 9 * 4?

The answer is 62084.

Congratulations! You've successfully solved this mathematical expression. Give yourself a pat on the back – you've earned it! We started with a seemingly complex problem, but by breaking it down into smaller, manageable steps and following the order of operations, we were able to find the solution. This is a testament to the power of systematic problem-solving. It's not just about the answer itself, but about the process we used to get there.

Think of it as climbing a mountain. The summit might seem far away and daunting at first, but by taking one step at a time, we gradually make progress and eventually reach the top. Similarly, in mathematics, we can conquer even the most challenging problems by approaching them methodically and with a clear understanding of the rules and principles involved. This experience will not only help you in your math studies but also in other areas of your life where problem-solving skills are essential.

Tips for Solving Similar Expressions

Now that we've conquered this expression, let's talk about some tips that can help you tackle similar problems in the future. These are like the tools in your mathematical toolbox, ready to be used whenever you encounter a new challenge.

  1. Always remember PEMDAS/BODMAS: This is your guiding star in the world of mathematical expressions. Make sure you always follow the order of operations to avoid making mistakes.
  2. Break it down: Complex expressions can be overwhelming, so break them down into smaller, more manageable steps. This makes the problem less intimidating and easier to solve.
  3. Double-check your work: It's always a good idea to double-check your calculations, especially in longer expressions. A small mistake early on can throw off the entire answer.
  4. Use a calculator (when allowed): Calculators are great tools for handling large numbers and complex calculations. Don't be afraid to use them, but make sure you still understand the underlying concepts.
  5. Practice makes perfect: The more you practice solving mathematical expressions, the better you'll become at it. So, keep practicing, and don't get discouraged if you make mistakes along the way.
  6. Stay organized: Keep your work neat and organized. This will help you keep track of your calculations and make it easier to spot any errors.
  7. Understand the concepts: It's not enough to just memorize the rules; you need to understand the underlying mathematical concepts. This will help you solve problems more effectively and apply your knowledge in different situations.

By keeping these tips in mind, you'll be well-equipped to tackle any mathematical expression that comes your way. Remember, math is like a muscle – the more you use it, the stronger it becomes!

Conclusion

So, there you have it! We've successfully solved the mathematical expression 56 + 108 + 20 * 86 * 9 * 4, and the answer is 62084. We walked through the entire process step by step, from understanding the order of operations to performing the multiplications and additions. We also shared some valuable tips for solving similar expressions in the future.

Remember, math isn't about memorizing formulas or getting the right answer every time. It's about understanding the process, developing problem-solving skills, and building confidence in your ability to tackle challenges. So, keep practicing, keep exploring, and never be afraid to ask questions. You've got this! And who knows, maybe you'll even start to enjoy the thrill of solving mathematical puzzles. Keep up the great work, guys!