Solve 7-9(17+7) ÷ 6²: A Step-by-Step Guide

by Omar Yusuf 43 views

Hey guys! Math can sometimes feel like navigating a maze, especially when you're faced with expressions that have multiple operations. But don't worry, we're going to break down a classic problem today: evaluating the expression 7-9(17+7) ÷ 6². This isn't just about getting the right answer; it's about understanding the fundamental order of operations that governs how we solve mathematical problems. Mastering this order is crucial, whether you're tackling algebra, calculus, or even everyday calculations.

The Order of Operations: PEMDAS/BODMAS

Before we dive into the problem, let's quickly review the order of operations, often remembered by the acronyms PEMDAS or BODMAS:

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

This order dictates the sequence in which we perform operations to ensure we arrive at the correct solution. Think of it as a set of traffic rules for the world of math, keeping everything flowing smoothly and consistently. Ignoring this order can lead to wildly incorrect results, so it's super important to get it right!

Why Does the Order of Operations Matter?

You might be wondering, why this specific order? Why not just go left to right? The reason is that the order of operations is designed to reflect the mathematical structure and relationships between operations. For example, exponents represent repeated multiplication, so they naturally come before simple multiplication or addition. Similarly, parentheses group operations together, indicating that those calculations should be performed as a single unit before anything else. Without a standardized order, mathematical expressions would be ambiguous and could have multiple interpretations, making it impossible to communicate mathematical ideas effectively.

Imagine trying to build a house without a blueprint – you might end up with a confusing mess! The order of operations is our blueprint for mathematical calculations, ensuring that everyone arrives at the same answer and understands the underlying logic.

Common Mistakes to Avoid

One of the biggest pitfalls in evaluating expressions is forgetting the order of operations. It's tempting to just work from left to right, but this will almost always lead to errors. Another common mistake is treating multiplication and division (or addition and subtraction) as having a strict order – remember, they are performed from left to right. For example, in the expression 10 ÷ 2 × 5, you should divide first (10 ÷ 2 = 5) and then multiply (5 × 5 = 25), not the other way around. Paying close attention to these nuances can make a big difference in your accuracy.

Step-by-Step Solution: 7-9(17+7) ÷ 6²

Okay, let's tackle our problem step-by-step, making sure to follow PEMDAS/BODMAS.

1. Parentheses/Brackets

Our expression is 7-9(17+7) ÷ 6². The first thing we need to do is simplify what's inside the parentheses:

17 + 7 = 24

So now our expression becomes:

7 - 9(24) ÷ 6²

2. Exponents/Orders

Next up are the exponents. We have 6², which means 6 raised to the power of 2 (6 multiplied by itself):

6² = 6 * 6 = 36

Our expression now looks like this:

7 - 9(24) ÷ 36

3. Multiplication and Division (from left to right)

Now we handle multiplication and division. Remember, we perform these operations from left to right. First, we have 9(24):

9 * 24 = 216

Our expression is now:

7 - 216 ÷ 36

Next, we perform the division:

216 ÷ 36 = 6

So our expression becomes:

7 - 6

4. Addition and Subtraction (from left to right)

Finally, we perform addition and subtraction, again from left to right. In this case, we only have one subtraction:

7 - 6 = 1

The Final Answer

Therefore, the value of the expression 7-9(17+7) ÷ 6² is 1. 🎉

Putting It All Together: A Recap

Let's quickly recap the steps we took to solve this problem:

  1. Parentheses: Simplified (17+7) to 24.
  2. Exponents: Evaluated 6² as 36.
  3. Multiplication: Multiplied 9 and 24 to get 216.
  4. Division: Divided 216 by 36 to get 6.
  5. Subtraction: Subtracted 6 from 7 to get the final answer, 1.

By following the order of operations diligently, we were able to break down a seemingly complex expression into a simple calculation. This systematic approach is key to success in mathematics.

Practice Makes Perfect: Try These Problems!

Now that you've seen how to tackle this problem, it's time to put your skills to the test! Here are a few similar expressions for you to evaluate. Remember to follow PEMDAS/BODMAS and work through each step carefully.

  1. 10 + 5 × 2 - 15 ÷ 3
  2. (8 - 2)² + 4 × 3
  3. 24 ÷ (2 × 3) + 16 - 4²

Working through these practice problems will solidify your understanding of the order of operations and build your confidence in tackling more complex mathematical challenges. Don't be afraid to make mistakes – they're a natural part of the learning process. The important thing is to learn from them and keep practicing!

Beyond the Basics: Real-World Applications

The order of operations isn't just a theoretical concept; it has practical applications in many areas of life. From programming computers to calculating finances, understanding the order of operations is essential for ensuring accuracy and consistency. For example, in programming, mathematical expressions are used extensively, and the order of operations determines how those expressions are evaluated. In finance, calculating interest rates or loan payments requires a precise understanding of the order in which operations should be performed.

Even in everyday situations, we often implicitly use the order of operations. Imagine you're baking a cake and the recipe says to add the dry ingredients before the wet ingredients and then bake at a certain temperature for a specific time. This is essentially an order of operations for cooking! Recognizing the importance of order in different contexts can help you appreciate the broader relevance of mathematical principles.

Conclusion: Mastering the Order of Operations

So, there you have it! We've successfully evaluated the expression 7-9(17+7) ÷ 6² by carefully following the order of operations. Remember, PEMDAS/BODMAS is your friend in the world of math. It provides a clear roadmap for solving expressions and ensures that everyone arrives at the same correct answer. Keep practicing, and you'll become a master of mathematical operations in no time! Keep an eye out for more math adventures, guys! You've got this!