Math Problem: Solve $13+[32 extdiv 4+(56+79)]$

Welcome to our math exploration for today! We're going to dive into a fascinating arithmetic problem that involves several operations: $13+[32 extdiv 4+(56+79)]$. This problem is a fantastic way to practice the order of operations, often remembered by the acronym PEMDAS or BODMAS. These rules are crucial in mathematics to ensure everyone arrives at the same, correct answer when evaluating expressions. Without a standardized order, ambiguous results would be common, leading to confusion in fields like science, engineering, and even everyday calculations. So, let's break down this expression step-by-step, ensuring we follow the rules meticulously. We'll start by looking at the innermost parentheses and work our way outwards, tackling division and addition as we go. Understanding how to correctly solve such problems not only strengthens your mathematical foundation but also hones your logical thinking and problem-solving skills. It's like unlocking a small puzzle, and the satisfaction of reaching the correct solution is quite rewarding. We’ll also discuss why each step is important and how it contributes to the final answer. This isn't just about getting the number; it's about understanding the process and the underlying mathematical principles that govern it. So, grab a pen and paper, and let's get started on unraveling this mathematical challenge together!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even touch the numbers in $13+[32 extdiv 4+(56+79)]$, it's essential to have a firm grasp on the order of operations. This is the universal language that mathematicians use to avoid confusion. You've likely encountered acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS (Brackets, Orders, Division and Multiplication from left to right, Addition and Subtraction from left to right). Both serve the same purpose: to provide a clear hierarchy for performing calculations. In our specific problem, we don't have exponents or orders (like square roots), but we do have parentheses, division, and addition. The first rule is to always tackle anything inside parentheses first. If there are nested parentheses, like in our expression, we start with the innermost set and work our way out. After dealing with parentheses, we move on to multiplication and division. These have equal priority, so we perform them as they appear from left to right. Finally, we handle addition and subtraction, which also have equal priority and are performed from left to right. This systematic approach ensures that every calculation yields a consistent and accurate result, regardless of who is performing it or where in the world they are. For instance, if we ignored the parentheses in $13+[32 extdiv 4+(56+79)]$ and just started from the left, we might do $13+32$ first, which would lead to a completely different and incorrect answer. The parentheses act as indicators, telling us which parts of the expression need to be resolved before others. They group numbers and operations, signaling that these should be treated as a single unit for the purpose of calculation. This concept is fundamental not just for solving this particular problem but for all of algebra and beyond. It's the bedrock upon which more complex mathematical structures are built. Therefore, mastering PEMDAS/BODMAS is a critical step in your mathematical journey.

Step-by-Step Solution for $13+[32 extdiv 4+(56+79)]$

Now, let's apply the order of operations to our problem: $13+[32 extdiv 4+(56+79)]$. We'll go through it methodically.

Step 1: Innermost Parentheses

The first thing we look for are the innermost parentheses. In our expression, that's $(56+79)$. Let's solve this addition:

$56 + 79 = 135$

Our expression now looks like this: $13+[32 extdiv 4+135]$.

Step 2: Next Level of Parentheses

We still have parentheses to deal with: $[32 extdiv 4+135]$. Inside these brackets, we have a division and an addition. According to PEMDAS/BODMAS, division comes before addition. So, we perform the division first:

$32 extdiv 4 = 8$

Now, the expression inside the brackets becomes $8+135$. Let's solve this addition:

$8 + 135 = 143$

Our expression has now been simplified to: $13+143$.

Step 3: Final Addition

We are left with a simple addition problem. This is the final step:

$13 + 143 = 156$

So, the final answer to the problem $13+[32 extdiv 4+(56+79)]$ is 156.

This step-by-step approach ensures that we address each part of the expression in the correct sequence, leading to the accurate result. It's a testament to the power and necessity of the order of operations. Each calculation, no matter how small, builds upon the previous one, guiding us towards the final solution. Imagine if we had done $4+56$ first. That would be $60$, then $32 extdiv 60$, which is a fraction. Then adding $79$ to that fraction, and then adding $13$. This would lead to a significantly different and incorrect answer. The structure provided by the parentheses and the rules of PEMDAS/BODMAS are what make mathematics a consistent and reliable discipline. It's like following a recipe; if you skip steps or use ingredients in the wrong order, the dish won't turn out as intended. Similarly, in mathematics, adhering to the order of operations is key to achieving the correct outcome. We started with the innermost operation, $56+79$, then moved to the operations within the next set of grouping symbols, performing the division $32 extdiv 4$ before the addition, and finally completed the outermost addition. This methodical progression is what guarantees the correctness of our final answer, 156.

Why is the Order of Operations Important?

The importance of the order of operations, like PEMDAS/BODMAS, cannot be overstated in the realm of mathematics and its applications. Consistency and accuracy are the cornerstones of any scientific or engineering endeavor, and mathematical expressions are the language used to describe these disciplines. If different individuals interpret the same mathematical expression in different ways, it could lead to catastrophic errors in calculations, designs, or scientific theories. For example, in computer programming, precise mathematical operations are fundamental. A slight misinterpretation of an expression due to incorrect order of operations could cause a program to malfunction or produce incorrect data. Similarly, in physics, when calculating forces, velocities, or trajectories, the exact sequence of operations is critical. Imagine calculating the trajectory of a projectile; if the addition and division were swapped, the predicted landing point could be wildly off, with significant real-world consequences. Furthermore, the order of operations facilitates communication and collaboration. When mathematicians, scientists, or engineers share their work, they rely on a common set of rules to ensure their equations are understood precisely as intended. This standardized approach allows for the efficient building upon previous discoveries and the collaborative development of complex theories. It transforms ambiguous mathematical statements into clear, unambiguous instructions. For $13+[32 extdiv 4+(56+79)]$, the parentheses clearly indicate which additions must be performed before the division and the final addition. The division has precedence over the addition within the brackets, and all these operations are performed before the initial $13$ is added. This structure is not arbitrary; it's a convention developed over centuries to bring order to mathematical expression. It allows for the construction of complex algebraic expressions and equations that can accurately model intricate real-world phenomena. Without this order, mathematics would descend into chaos, with no single correct answer to any given problem. It's the invisible framework that supports all quantitative reasoning, making it a reliable tool for understanding and manipulating the world around us. It provides a predictable and repeatable method for evaluation, ensuring that results are not based on guesswork or personal preference, but on universally accepted rules.

Conclusion

In conclusion, solving the mathematical expression $13+[32 extdiv 4+(56+79)]$ requires a strict adherence to the order of operations, PEMDAS/BODMAS. By meticulously following the steps—starting with the innermost parentheses, then addressing operations within the next set of brackets by prioritizing division over addition, and finally performing the last addition—we arrived at the correct answer of 156. This problem serves as a clear illustration of why these rules are not just arbitrary conventions but essential tools for ensuring consistency, accuracy, and clear communication in mathematics. Mastering the order of operations is a fundamental skill that underpins success in more advanced mathematical studies and in many scientific and technical fields. It allows us to break down complex problems into manageable parts and solve them with confidence. The ability to correctly interpret and evaluate mathematical expressions is a powerful skill that enhances logical thinking and problem-solving capabilities.

For further exploration into the fundamental principles of mathematics and arithmetic operations, you can visit Khan Academy. They offer a wealth of resources, including detailed explanations and practice exercises on the order of operations and other essential math topics.