Solving [(12 + 18 ÷ 3) * 2 - 5]² ÷ 25 A Step-by-Step Guide

Hey guys! Math can sometimes feel like navigating a maze, right? But don't worry, we're here to break down even the trickiest problems into bite-sized, easy-to-follow steps. Today, we're tackling the expression [(12 + 18 ÷ 3) * 2 - 5]² ÷ 25. This might look intimidating at first glance, but with the order of operations (PEMDAS/BODMAS) as our trusty guide, we'll solve it together. So, grab your pencils and let's dive in!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we jump into the problem, it's crucial to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS. This is the golden rule of mathematical calculations, ensuring we all arrive at the same answer. Let's break it down:

• Parentheses / Brackets: First, we tackle anything inside parentheses or brackets.
• Exponents / Orders: Next up are exponents (powers) or orders (like square roots).
• Multiplication and Division: These are done from left to right.
• Addition and Subtraction: Finally, addition and subtraction are performed from left to right.

Keeping this order in mind is key to accurately solving any mathematical expression. It's like having a roadmap for our calculations, ensuring we don't take any wrong turns. Think of it as the grammar of mathematics; it provides the structure and clarity we need to communicate effectively with numbers. Without this order, we'd be left with a jumbled mess of operations, potentially leading to incorrect results.

Now, let's see how this applies to our problem. We have parentheses, division, multiplication, subtraction, an exponent, and then division again. Following PEMDAS/BODMAS, we know exactly where to start and how to proceed. This structured approach is what makes even complex problems manageable. We're not just throwing numbers around randomly; we're following a logical sequence that guarantees the correct solution. So, with our roadmap in hand, let's start unraveling this mathematical puzzle, step by logical step.

Step 1: Tackling the Parentheses (12 + 18 ÷ 3)

The first stop on our journey is the parentheses: (12 + 18 ÷ 3). Inside the parentheses, we have both addition and division. According to PEMDAS/BODMAS, division takes precedence. So, we'll handle the 18 ÷ 3 part first. 18 divided by 3 is 6. This simplifies our expression within the parentheses to (12 + 6).

Now, we can easily add 12 and 6, which gives us 18. So, the entire expression within the parentheses simplifies to 18. This might seem like a small victory, but it's a significant step in simplifying the overall problem. We've effectively condensed a more complex operation into a single number, making the next steps much easier to handle. Remember, breaking down a large problem into smaller, manageable chunks is a key strategy in mathematics (and in life!).

Think of it like decluttering a room. You wouldn't try to organize everything at once. Instead, you'd focus on one area, one drawer, or even one item at a time. Similarly, in math, we focus on one operation at a time, following the prescribed order. By prioritizing division before addition within the parentheses, we're ensuring that we're adhering to the fundamental rules of arithmetic. This meticulous approach not only guarantees accuracy but also builds a solid foundation for tackling more advanced mathematical concepts in the future.

Step 2: Multiplication (18 * 2)

With the parentheses neatly resolved, our expression now looks like this: [18 * 2 - 5]² ÷ 25. Moving along our PEMDAS/BODMAS roadmap, we encounter multiplication. We have 18 multiplied by 2, which equals 36. This further simplifies our expression to [36 - 5]² ÷ 25.

Multiplication is a fundamental operation, and in this context, it's bridging the gap between the simplified parentheses and the next operation, which is subtraction. It's like building a bridge across a river, connecting two previously separate entities. By performing the multiplication, we're one step closer to isolating the core value within the brackets. This step highlights the interconnectedness of mathematical operations; each one builds upon the previous, leading us towards the final solution.

The importance of accurate multiplication cannot be overstated. A mistake here would throw off the entire calculation. That's why it's crucial to double-check our work and ensure we're handling each step with care. Think of it as laying the foundation for a building; if the foundation is weak, the entire structure is at risk. Similarly, in math, each operation is a building block, and a solid understanding of multiplication is essential for constructing accurate solutions.

Step 3: Subtraction (36 - 5)

Now, let's tackle the subtraction within the brackets: [36 - 5]² ÷ 25. Subtracting 5 from 36 gives us 31. Our expression is now beautifully streamlined to 31² ÷ 25. We're making excellent progress! This step exemplifies the power of simplification in mathematics. By methodically working through each operation, we've transformed a seemingly complex expression into a much more manageable form. This is a testament to the elegance and efficiency of the order of operations.

Subtraction, in this context, is the final operation within the brackets, effectively isolating the value that will be squared in the next step. It's like the last piece of a puzzle falling into place, revealing a clearer picture. By accurately performing the subtraction, we're setting the stage for the exponent, which will significantly impact the final result. This underscores the importance of precision in each step; a small error here could have a cascading effect on the rest of the calculation.

Remember, math is not just about getting the right answer; it's about understanding the process and the relationships between different operations. This step highlights the importance of subtraction as a fundamental arithmetic operation and its role in simplifying expressions. By mastering subtraction, we're not only solving this particular problem but also building a stronger foundation for future mathematical challenges.

Step 4: Exponent (31²)

Time for the exponent! We have 31² ÷ 25. Remember, the exponent tells us to multiply the base (31 in this case) by itself. So, 31² means 31 * 31, which equals 961. Our expression is now a sleek 961 ÷ 25. We're in the home stretch now! This step showcases the power of exponents in mathematics. Squaring a number can significantly alter its value, and in this case, it's transformed 31 into a much larger number, 961.

Exponents are a fundamental concept in mathematics, used extensively in algebra, calculus, and various other branches. Understanding how exponents work is crucial for solving a wide range of problems, from simple calculations to complex equations. In this context, the exponent plays a pivotal role in determining the magnitude of the final result. It's a reminder that mathematical operations are not always linear; exponents introduce a non-linear element that can lead to significant changes in value.

The calculation of 31² might seem straightforward, but it's essential to ensure accuracy. A small error in multiplication here would lead to a completely different final answer. This highlights the importance of meticulous calculation and the value of double-checking our work. By correctly applying the exponent, we're one step closer to the final solution, demonstrating the power and precision of mathematical operations.

Step 5: Final Division (961 ÷ 25)

Finally, we arrive at the last step: 961 ÷ 25. Performing this division gives us 38.44. And there you have it! The solution to the expression [(12 + 18 ÷ 3) * 2 - 5]² ÷ 25 is 38.44. Woohoo! We did it! This final division brings the entire problem to a satisfying conclusion. It's the culmination of all the previous steps, demonstrating how each operation contributes to the overall solution.

Division is a fundamental arithmetic operation, and in this context, it's the final step in scaling down the value to its ultimate form. It's like the final polish on a finished product, bringing clarity and precision to the result. By accurately performing the division, we're ensuring that our answer is not only correct but also presented in its most concise form. This step underscores the importance of division in mathematics and its role in distributing quantities and determining proportions.

The result, 38.44, is a decimal number, which is a common occurrence in mathematical calculations. It's a reminder that math is not always about whole numbers; decimal numbers play a crucial role in representing fractional parts and providing more precise measurements. Understanding decimal numbers is essential for a comprehensive understanding of mathematics. By arriving at this decimal solution, we've successfully navigated the entire expression, demonstrating our mastery of the order of operations and our ability to perform various arithmetic operations with accuracy and confidence.

Conclusion: Mastering the Order of Operations

So, there you have it! We've successfully solved the expression [(12 + 18 ÷ 3) * 2 - 5]² ÷ 25, arriving at the answer of 38.44. The key to conquering this problem, and many others in mathematics, is understanding and applying the order of operations (PEMDAS/BODMAS). It's like having a secret code that unlocks the solution, ensuring we tackle each operation in the correct sequence.

Math might seem daunting at times, but by breaking down complex problems into smaller, manageable steps, we can approach them with confidence and clarity. Remember, each operation builds upon the previous one, and a solid understanding of the fundamentals is crucial for success. Whether it's parentheses, exponents, multiplication, division, addition, or subtraction, each operation has its place in the grand scheme of mathematical calculations. By mastering these operations and the order in which they're performed, we're not just solving problems; we're building a strong foundation for future mathematical endeavors.

Keep practicing, keep exploring, and never be afraid to ask questions. Math is a journey, and every problem solved is a step forward. So, keep that calculator handy, and let's continue our mathematical adventures together!