Order Of Operations First Step Evaluating (-4)^2+6 ÷(-3+4)(2)-5

Hey guys! Let's break down this math problem step by step. We're tackling the expression $(-4)^2+6 \div(-3+4)(2)-5$ using the order of operations, and the big question is: what do we do first? To nail this, we need to remember our trusty PEMDAS (or BODMAS, if you prefer):

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

Understanding PEMDAS/BODMAS

Think of PEMDAS as your roadmap for solving mathematical expressions. It tells you exactly what to handle first, second, and so on, ensuring everyone arrives at the same correct answer. Without a consistent order, math would be chaotic! So, let’s dive deeper into each part of PEMDAS to see how it guides us. Understanding this order is super crucial, and we will explore each step thoroughly.

1. Parentheses/Brackets: The First Priority

Parentheses (or brackets, depending on where you learned your math!) are like VIP sections in an expression. Anything inside them gets top priority. This is where you start simplifying, no matter what else is going on in the equation. Parentheses might contain simple operations like addition or subtraction, or they could house more complex calculations. The key is to treat whatever is inside them as a mini-problem to be solved before you tackle the rest.

For instance, if you see something like 2 * (3 + 4), you must add 3 and 4 first, which gives you 7. Then, you can multiply 2 by 7. Skipping this step would throw off your entire calculation. This initial focus ensures that you're working with the most fundamental parts of the expression first, setting the stage for accurate calculations down the line. Remember, parentheses aren't just about grouping; they're about dictating the order of operations to maintain mathematical integrity.

2. Exponents/Orders: Powering Through

After you've handled the parentheses, exponents come next in line. Exponents (also known as orders or powers) tell you how many times a number is multiplied by itself. They're written as a small number raised to the right of the base number. For example, in 5^2, the 2 is the exponent, and it means you multiply 5 by itself (5 * 5).

Dealing with exponents early is crucial because they can significantly change the value of a term. Consider the difference between 3 * 2^2 and (3 * 2)^2. In the first case, you calculate 2^2 (which is 4) and then multiply by 3, giving you 12. In the second case, you multiply 3 and 2 first, getting 6, and then square it, resulting in 36. See how different the outcomes are? Getting exponents right early on prevents these kinds of errors and keeps your calculations on track. This step ensures that the powers and roots are correctly evaluated before moving on to multiplication, division, addition, or subtraction, maintaining the proper mathematical hierarchy.

3. Multiplication and Division: A Left-to-Right Affair

Once you've taken care of parentheses and exponents, it's time for multiplication and division. Now, here's a neat little trick to remember: these operations have equal priority. This means you don't always do multiplication before division; instead, you work from left to right, tackling whichever one comes first in the expression.

Imagine you're reading a sentence – you process words in the order they appear, right? It's the same with multiplication and division. For example, in the expression 10 / 2 * 3, you would first divide 10 by 2 (which gives you 5) and then multiply by 3, ending up with 15. If you multiplied first, you'd get a completely different answer, highlighting why this left-to-right rule is so important. This approach guarantees that the operations are performed in the correct sequence, respecting their equal precedence and leading to the accurate result.

4. Addition and Subtraction: The Final Touches

Last but not least, we have addition and subtraction. Just like multiplication and division, these operations have equal priority, so you handle them from left to right. By this stage, you've simplified most of the expression, and what's left is usually straightforward.

Think of it as the final polish on your mathematical masterpiece. For instance, in 8 + 5 - 2, you first add 8 and 5 to get 13, and then subtract 2, leaving you with 11. Again, going from left to right ensures accuracy. It’s a simple rule, but sticking to it prevents errors and ensures that your final answer is spot on. This final step consolidates all previous calculations, providing the definitive solution to the expression.

Applying PEMDAS to Our Problem

Okay, let's use PEMDAS to solve our specific problem: $(-4)^2+6 \div(-3+4)(2)-5$.

1. Parentheses: We see $(-3+4)$ inside parentheses. So, we add $-3$ and $4$, which equals $1$. Our expression now looks like this: $(-4)^2+6 \div(1)(2)-5$.
2. Exponents: Next up is the exponent. We have $(-4)^2$, which means $-4$ multiplied by itself. $-4 * -4 = 16$. The expression becomes: $16+6 \div(1)(2)-5$.
3. Multiplication and Division: Here, we tackle division and multiplication from left to right. We have $6 \div(1)$, which is $6$. Then, we multiply $6$ by $2$, which is $12$. Our expression now reads: $16 + 12 - 5$.
4. Addition and Subtraction: Finally, we do addition and subtraction from left to right. $16 + 12 = 28$, and then $28 - 5 = 23$.

So, the final answer is $23$. But our main question was, what's the first step? Based on PEMDAS, it’s the parentheses.

Breaking Down the Options

Let's look at the answer choices provided:

A. Subtract 5 from 2. B. Divide 6 by -3. C. Add -3 and 4. D. Multiply 4 and 2.

Which one lines up with our first step according to PEMDAS?

• Option A, "Subtract 5 from 2," is incorrect. Subtraction comes much later in the order of operations.
• Option B, "Divide 6 by -3," might seem tempting, but remember we need to handle parentheses first.
• Option C, "Add -3 and 4," is the correct answer! This is the operation inside the parentheses, making it our top priority.
• Option D, "Multiply 4 and 2," is also incorrect because we address parentheses and exponents before multiplication.

Conclusion

So, the first step in evaluating the expression $(-4)^2+6 \div(-3+4)(2)-5$ using the order of operations is to C. Add -3 and 4. Remember, PEMDAS is your friend! Stick to the order, and you'll conquer any mathematical expression that comes your way. Keep practicing, guys, and you'll become math whizzes in no time! Understanding and applying the order of operations is a fundamental skill in mathematics, essential for tackling more complex problems with confidence and accuracy. Great job, and keep up the awesome work!