Solving Mathematical Expressions A Step-by-Step Guide For [(17-15)² + (7-12)²] - [(6-7) (12-23)]

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Hey guys! Today, we’re diving into a mathematical expression that might look a bit intimidating at first glance, but trust me, it’s totally manageable when we break it down step by step. We're going to solve: [(17-15)² + (7-12)²] - [(6-7) (12-23)]. I’ll walk you through each operation, explaining the logic and the order in which we tackle things. So, grab your calculators (or your mental math muscles!) and let’s get started!

Understanding the Order of Operations

Before we even think about plugging in numbers, it’s super important to understand the order of operations. Remember PEMDAS or BODMAS? It’s the golden rule for solving mathematical expressions. It tells us the sequence in which we should perform operations:

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

Following this order ensures we all arrive at the same correct answer. If we jumped around randomly, we’d end up with a mathematical mess! Seriously, this is the key. Keep PEMDAS or BODMAS in your mind as we go through the problem.

Breaking Down the Expression

Okay, let's rewrite the expression here so we can easily reference it: [(17-15)² + (7-12)²] - [(6-7) (12-23)]. We've got parentheses, exponents, subtraction, and multiplication. Time to put PEMDAS/BODMAS into action!

We will start by addressing the parentheses first. This is where we'll simplify the expressions inside each set of brackets before moving on to exponents or anything else. Think of it as clearing the path for the more complex operations that follow.

Step 1: Solving the Innermost Parentheses

Our expression has several sets of parentheses, so we’ll start with the innermost ones. Let's break it down:

  • (17 - 15): This is a simple subtraction. 17 minus 15 equals 2. So, we replace (17 - 15) with 2.
  • (7 - 12): Here, we're subtracting a larger number from a smaller one, which will give us a negative result. 7 minus 12 equals -5.
  • (6 - 7): Again, we subtract a larger number from a smaller one. 6 minus 7 equals -1.
  • (12 - 23): Another subtraction with a negative result. 12 minus 23 equals -11.

Now, let's rewrite the expression with these simplifications: [2² + (-5)²] - [(-1) (-11)]. See how much cleaner it looks already? We've knocked out the basic subtractions within the parentheses. This is a crucial step because it simplifies the expression and sets us up for the next operations.

Step 2: Dealing with the Exponents

Next up, we tackle the exponents. Remember, an exponent tells us how many times to multiply a number by itself. In our simplified expression, [2² + (-5)²] - [(-1) (-11)], we have two exponents to deal with:

  • 2²: This means 2 multiplied by itself, which is 2 * 2 = 4.
  • (-5)²: This means -5 multiplied by itself, which is (-5) * (-5) = 25. Remember, a negative number multiplied by a negative number gives a positive result.

Let's replace these exponents in our expression: [4 + 25] - [(-1) (-11)]. We're making great progress! The exponents are gone, and we're left with simpler operations.

Understanding exponents is crucial not just for this problem, but for many areas of math. They show up in algebra, geometry, and even calculus. So, mastering this step is a big win!

Step 3: Multiplication within Parentheses

Now, let’s focus on the remaining parentheses. We have [4 + 25] - [(-1) (-11)]. Within the second set of brackets, we have a multiplication operation: (-1) * (-11).

  • (-1) * (-11): A negative number multiplied by a negative number gives a positive result. So, -1 times -11 equals 11.

Replace this multiplication in our expression: [4 + 25] - [11]. We've simplified the multiplication, and the expression is looking even cleaner.

Multiplication is a fundamental operation, and being comfortable with multiplying positive and negative numbers is key to avoiding mistakes. Pay close attention to the signs, and you'll be golden!

Step 4: Addition within Parentheses

We're almost there! Let’s continue simplifying within the parentheses. We now have [4 + 25] - [11]. The first set of brackets contains an addition operation:

  • (4 + 25): This is a simple addition. 4 plus 25 equals 29.

Replace this addition in our expression: [29] - [11]. Now, we have single numbers within each set of brackets, making the final step straightforward.

Addition, like multiplication, is a basic operation that forms the foundation for more complex math. Make sure you're comfortable with adding numbers of all sizes, and you'll be well-prepared for any mathematical challenge!

Step 5: The Final Subtraction

Finally, we arrive at the last operation: subtraction. Our expression is now [29] - [11]. This is a simple subtraction between two numbers:

  • 29 - 11: 29 minus 11 equals 18.

So, the final answer to our expression is 18! We did it! We started with a seemingly complex expression and, by following the order of operations and breaking it down step by step, we arrived at the solution.

Checking Our Work (Always a Good Idea!)

Before we celebrate too much, it’s always a good idea to double-check our work. Math mistakes can happen, and it's better to catch them ourselves than to leave them lurking in our calculations.

Let's quickly recap our steps:

  1. Solved the innermost parentheses: (17-15) = 2, (7-12) = -5, (6-7) = -1, (12-23) = -11
  2. Simplified the expression: [2² + (-5)²] - [(-1) (-11)]
  3. Dealt with exponents: 2² = 4, (-5)² = 25
  4. Simplified the expression: [4 + 25] - [(-1) (-11)]
  5. Performed multiplication: (-1) * (-11) = 11
  6. Simplified the expression: [4 + 25] - [11]
  7. Performed addition: 4 + 25 = 29
  8. Simplified the expression: [29] - [11]
  9. Performed the final subtraction: 29 - 11 = 18

Our steps look good, and the logic is sound. We can be confident in our answer!

Conclusion: Mastering the Art of Mathematical Expressions

So, guys, we’ve successfully navigated a somewhat complex mathematical expression. Remember, the key is to break things down into manageable steps and follow the order of operations. By understanding PEMDAS/BODMAS and working through each step carefully, you can tackle any expression that comes your way.

Math might seem intimidating sometimes, but with practice and a systematic approach, you’ll find it’s not so scary after all. Keep practicing, keep exploring, and most importantly, keep asking questions! You've got this!

If you have any more mathematical expressions you’d like to solve, or if you want to dive deeper into specific concepts, let me know. Let's keep learning and growing together! Keep an eye out for more math adventures, and until next time, happy calculating!