First Operation: Simplifying 10 + (2 ⋅ 3² - 4 ⋅ 2)

Hey everyone! Today, we're diving into a fundamental concept in mathematics: the order of operations. It's like the grammar rules of math – you need to follow them to get the right answer. We'll be dissecting the expression $10 + (2 ⋅ 3^2 - 4 ⋅ 2)$ to figure out which operation comes first. This isn't just about getting the correct answer to this specific problem; it's about understanding the underlying principles that govern how we approach mathematical expressions. So, grab your thinking caps, and let's get started!

The Order of Operations: PEMDAS/BODMAS

Before we jump into the problem, let's quickly recap the order of operations. You might have heard of the acronyms PEMDAS or BODMAS, which are handy ways to remember the sequence:

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

Think of it as a hierarchy. Operations at the top of the list take precedence over those at the bottom. This ensures that everyone solves the same problem in the same way, leading to a consistent and correct answer. Without a standardized order, math would be chaos!

Breaking Down the Expression: 10 + (2 ⋅ 3² - 4 ⋅ 2)

Now, let's apply this to our expression: $10 + (2 ⋅ 3^2 - 4 ⋅ 2)$. The first thing we notice is the parentheses. According to PEMDAS/BODMAS, we need to tackle everything inside the parentheses before we can do anything else. The parentheses act like a VIP section in a math problem – they get priority access.

Inside the parentheses, we have a mix of operations: multiplication, subtraction, and an exponent. This is where the order within the order comes into play! We need to follow PEMDAS/BODMAS within the parentheses as well. So, what's the first operation we encounter inside the parentheses?

According to the mnemonic, exponents come before multiplication, subtraction, addition, and division. Our expression inside the parentheses is $(2 ⋅ 3^2 - 4 ⋅ 2)$. The term $3^2$, or 3 squared, is an exponent. Therefore, the very first step we must take when simplifying this expression is to evaluate $3^2$.

Let's elaborate on why this is crucial. Exponents represent repeated multiplication. In this case, $3^2$ means $3 * 3$, which equals 9. If we didn't handle the exponent first, we might be tempted to multiply 2 by 3 before squaring, leading to a completely different result. Math is a precise language, and the order of operations is its grammar.

Why 3² is the First Step: A Detailed Explanation

Let's walk through the reasoning step-by-step to solidify our understanding. We've established that we need to work inside the parentheses first. Inside the parentheses, we have $2 ⋅ 3^2 - 4 ⋅ 2$. Now, let's consider the possible operations and their precedence:

1. Exponents: We have $3^2$, which needs to be evaluated.
2. Multiplication: We have two multiplication operations: $2 ⋅ 3^2$ and $4 ⋅ 2$.
3. Subtraction: We have one subtraction operation: the subtraction between the results of the two multiplication operations.

Following PEMDAS/BODMAS, exponents come before multiplication and subtraction. Therefore, $3^2$ is the operation we perform first.

To further illustrate this, imagine we ignored the order of operations and performed the multiplication $2 * 3$ first. We'd get 6, and then we'd have $6^2$, which is 36. This is drastically different from the correct value of $3^2$, which is 9. This simple example highlights the importance of adhering to the order of operations to maintain mathematical consistency and accuracy.

Continuing the Simplification

Although the question only asks for the first operation, let's briefly outline the next steps to give you a complete picture. Once we evaluate $3^2$ as 9, our expression within the parentheses becomes $(2 ⋅ 9 - 4 ⋅ 2)$.

Now, we have two multiplication operations: $2 ⋅ 9$ and $4 ⋅ 2$. According to PEMDAS/BODMAS, we perform multiplication and division from left to right. So, we'd calculate $2 ⋅ 9$ first, which is 18, and then $4 ⋅ 2$, which is 8. Our expression now looks like $(18 - 8)$.

Next, we perform the subtraction: $18 - 8 = 10$. So, the expression inside the parentheses simplifies to 10. Now, we have $10 + 10$, which is a simple addition, resulting in 20. Therefore, the simplified value of the original expression is 20.

Why the Other Options are Incorrect

Let's quickly address why the other answer choices are incorrect:

• (B) 2 ⋅ 3: While multiplication is present in the expression, the exponent $3^2$ takes precedence according to PEMDAS/BODMAS.
• (C) 4 ⋅ 2: This multiplication is also present, but it comes after the exponent in the order of operations.
• (D) 10 + 2: This addition is outside the parentheses and is the very last operation we would perform, not the first.

Choosing the correct operation isn't just about memorizing the rules; it's about understanding the mathematical logic behind them. Each step in the order of operations is designed to maintain consistency and ensure accurate results.

Conclusion: Mastering the Order of Operations

So, the answer to the question "Which operation is performed first when simplifying $10 + (2 ⋅ 3^2 - 4 ⋅ 2)$?" is definitively (A) 3². Understanding and applying the order of operations is a cornerstone of mathematics. It's a skill that will serve you well in algebra, calculus, and beyond. By remembering PEMDAS/BODMAS and practicing consistently, you'll become a master of simplifying expressions!

Remember, math is like a puzzle, and the order of operations is the key to unlocking it. Keep practicing, keep exploring, and you'll find that math can be both challenging and incredibly rewarding. Keep your fundamentals strong, and you'll be set up for mathematical success.