Solving $6-4 \div 2+7 \times(-\sqrt{9})$ A Step-by-Step Guide

Hey guys! Let's dive into a fun math problem that'll help us sharpen our skills in the order of operations. We're going to break down the expression $6-4 \div 2+7 \times(-\sqrt{9})$ step by step, so you'll not only get the answer but also understand why we solve it the way we do. Get ready to become a master of mathematical operations!

Understanding the Order of Operations

Before we jump into the problem, it's super important to understand the rules of the game. In mathematics, we have a specific order we need to follow when we're dealing with expressions that have multiple operations. This order is often remembered by the acronym PEMDAS, which stands for:

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

Think of PEMDAS as your trusty guide in the world of mathematical expressions. It ensures that we all get to the same correct answer, no matter who's solving the problem. Without this order, math would be chaotic, and we'd have a bunch of different answers floating around! So, let's keep PEMDAS in our toolkit as we tackle this problem.

The reason PEMDAS is so crucial is that it provides a standardized approach to solving mathematical expressions. Imagine if everyone could choose their own order of operations – we'd have a mathematical free-for-all! By adhering to PEMDAS, we maintain consistency and clarity in our calculations. Parentheses come first because they group terms together, indicating that these operations should be prioritized. Exponents follow, representing repeated multiplication and holding a higher precedence than basic multiplication and division. Multiplication and division are on the same level, and we work them from left to right, just like reading a sentence. This is also true for addition and subtraction, which come last in the order. This left-to-right approach ensures that expressions are evaluated unambiguously.

Moreover, understanding and applying the order of operations correctly is not just about getting the right answer in a textbook problem. It's a fundamental skill that underlies many areas of mathematics and science. From algebra to calculus, from physics to engineering, the principles of PEMDAS are constantly in play. A solid grasp of the order of operations enables us to simplify complex equations, solve for unknowns, and make accurate predictions in various fields. It's a cornerstone of mathematical literacy, empowering us to tackle a wide range of problems with confidence and precision.

Think of it like this: if you were building a house, you wouldn't start by putting up the roof before laying the foundation, right? Similarly, in math, we need to follow a logical sequence to arrive at the correct solution. PEMDAS is our blueprint for mathematical construction, ensuring that we build our answers on solid ground. So, whether you're balancing your budget, calculating the trajectory of a rocket, or just trying to figure out a tip at a restaurant, the order of operations will be your reliable companion. Mastering PEMDAS is not just about memorizing an acronym; it's about developing a structured, logical approach to problem-solving that will serve you well in all aspects of life.

Breaking Down the Problem: $6-4 \div 2+7 \times(-\sqrt{9})$

Okay, now that we've got our PEMDAS glasses on, let's break down the expression $6-4 \div 2+7 \times(-\sqrt{9})$. We're going to take it one step at a time, just like a detective solving a case. Our goal is to simplify the expression by following the correct order of operations.

Step 1: Parentheses and Roots

First up, we need to look for any parentheses or brackets. In our expression, we don't have any direct parentheses, but we do have something similar: a square root! The term $-\sqrt{9}$ acts a bit like a parenthetical expression because we need to simplify the square root before we can proceed with other operations. So, let's tackle that first.

The square root of 9 is 3, because 3 multiplied by itself (3 * 3) equals 9. Therefore, $-\sqrt{9}$ simplifies to -3. Now we can rewrite our expression as:

$6 - 4 \div 2 + 7 \times (-3)$

We've taken care of the root, which is like handling the first clue in our math mystery. By simplifying the square root, we've made the expression a little less complex and brought ourselves closer to the final answer. This step highlights the importance of recognizing and addressing any embedded operations, like roots, before moving on to the main arithmetic. It's like setting the stage for the rest of the calculation, ensuring that we're working with the simplest possible terms.

Step 2: Division

Next on our PEMDAS checklist is division. Looking at our updated expression, $6 - 4 \div 2 + 7 \times (-3)$, we see a division operation: $4 \div 2$. Let's perform this division:

$4 \div 2 = 2$

Now we can substitute this result back into our expression:

$6 - 2 + 7 \times (-3)$

By completing the division, we've further simplified the expression. We're systematically working through the operations, following the PEMDAS guide, and making progress towards our solution. Division, along with multiplication, holds a higher precedence than addition and subtraction, so it's crucial to address it at this stage. This step demonstrates the importance of prioritizing operations correctly to maintain the integrity of the mathematical expression.

Step 3: Multiplication

Following PEMDAS, we move on to multiplication. In our expression, $6 - 2 + 7 \times (-3)$, we have a multiplication operation: $7 \times (-3)$. Let's take care of that:

$7 \times (-3) = -21$

Now we substitute this result back into our expression:

$6 - 2 + (-21)$

We've tackled the multiplication, and our expression is getting even simpler. Notice how we're keeping track of the negative sign – that's super important in math! By performing the multiplication, we're one step closer to the final answer. This step reinforces the significance of handling multiplication (and division) before moving on to addition and subtraction, as dictated by the order of operations.

Step 4: Addition and Subtraction

Alright, we're in the home stretch! Now we just have addition and subtraction left. Remember, we perform these operations from left to right. Our expression is $6 - 2 + (-21)$.

First, let's do the subtraction: $6 - 2 = 4$

Now we have:

$4 + (-21)$

Next, let's do the addition: $4 + (-21) = -17$

And there we have it! We've simplified the entire expression.

The Final Answer

So, after carefully following the order of operations, we've found that:

$6-4 \div 2+7 \times(-\sqrt{9}) = -17$

Why This Matters: Real-World Applications

You might be thinking,