Evaluating The Expression 3 ⋅ (2-1) / (-10)^(-3) + 12 A Step-by-Step Guide
Evaluating mathematical expressions can sometimes feel like navigating a complex maze, but with a systematic approach and a solid understanding of the order of operations, even the most intricate expressions can be simplified. In this article, we will dissect the expression 3 ⋅ (2-1) / (-10)^(-3) + 12, providing a detailed, step-by-step solution to arrive at the correct answer. We'll not only solve the problem but also discuss the underlying mathematical principles and common pitfalls to avoid. This comprehensive guide aims to empower you with the knowledge and skills needed to confidently tackle similar mathematical challenges.
Understanding the Order of Operations
Before we dive into the specifics of the expression, it's crucial to reiterate the fundamental principle guiding mathematical evaluations: the order of operations. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this hierarchy dictates the sequence in which operations must be performed to ensure a consistent and accurate result.
- Parentheses: Operations within parentheses (or brackets) are always performed first. This step is crucial for isolating and simplifying terms within the expression.
- Exponents: Next, we handle exponents and roots. Understanding how to work with powers, including negative exponents, is essential for this step.
- Multiplication and Division: Multiplication and division are performed from left to right. It's important to note that these operations have equal precedence, so their order is determined by their position in the expression.
- Addition and Subtraction: Finally, addition and subtraction are performed from left to right, similar to multiplication and division.
By adhering to PEMDAS, we can ensure that our calculations are consistent and lead to the correct solution. Neglecting this order can result in significant errors, highlighting the importance of mastering this fundamental principle.
Step-by-Step Evaluation of the Expression
Now, let's apply the order of operations to the expression 3 ⋅ (2-1) / (-10)^(-3) + 12. We'll break down each step in detail, providing clear explanations and justifications for our actions.
Step 1: Simplify the Parentheses
The first step, according to PEMDAS, is to simplify the expression within the parentheses: (2 - 1). This is a straightforward subtraction:
(2 - 1) = 1
Now, our expression becomes:
3 ⋅ 1 / (-10)^(-3) + 12
This simplification sets the stage for the next operation, making the expression more manageable.
Step 2: Evaluate the Exponent
The next step involves dealing with the exponent: (-10)^(-3). A negative exponent indicates a reciprocal, meaning we need to take the reciprocal of the base raised to the positive exponent. In other words:
(-10)^(-3) = 1 / (-10)^3
Now, we need to calculate (-10)^3, which means multiplying -10 by itself three times:
(-10)^3 = (-10) ⋅ (-10) ⋅ (-10) = -1000
Therefore:
(-10)^(-3) = 1 / (-1000) = -0.001
Our expression now looks like this:
3 ⋅ 1 / (-0.001) + 12
Step 3: Perform Multiplication and Division
Following PEMDAS, we perform multiplication and division from left to right. First, we have the multiplication: 3 ⋅ 1:
3 ⋅ 1 = 3
Our expression is now:
3 / (-0.001) + 12
Next, we perform the division: 3 / (-0.001). Dividing by a small decimal can be tricky, so it's helpful to think of it as multiplying by the reciprocal. The reciprocal of -0.001 is -1000:
3 / (-0.001) = 3 ⋅ (-1000) = -3000
Now, the expression is simplified to:
-3000 + 12
Step 4: Perform Addition
The final step is the addition: -3000 + 12:
-3000 + 12 = -2988
Therefore, the final result of the expression is -2988.
The Correct Answer and Common Mistakes
Based on our step-by-step evaluation, the correct answer is B. (-2,988).
It's important to note that errors in evaluating expressions often stem from neglecting the order of operations. For example, someone might incorrectly perform the addition before the division, leading to a wrong result. Another common mistake is mishandling negative exponents, forgetting that they indicate a reciprocal. By carefully following PEMDAS and paying close attention to the rules of exponents, you can avoid these pitfalls.
Practice and Mastery
Mastering the evaluation of mathematical expressions requires consistent practice and a thorough understanding of the underlying principles. Work through various examples, focusing on applying PEMDAS correctly and paying attention to details such as negative signs and exponents. With each problem you solve, you'll build confidence and sharpen your skills. Consider exploring additional resources, such as online tutorials and practice exercises, to further enhance your understanding.
Conclusion
Evaluating the expression 3 ⋅ (2-1) / (-10)^(-3) + 12 demonstrates the importance of adhering to the order of operations and applying mathematical principles systematically. By breaking down the problem into manageable steps and carefully executing each operation, we arrived at the correct answer: -2988. This process highlights the power of structured problem-solving and the critical role of PEMDAS in mathematical evaluations. Remember, consistent practice and a solid understanding of the fundamentals are key to mastering mathematical expressions and achieving success in your mathematical endeavors.
This detailed guide provides not only the solution but also a comprehensive explanation of the steps involved, empowering you to tackle similar challenges with confidence. Keep practicing, stay focused, and you'll become proficient in evaluating even the most complex mathematical expressions.
Let's recap the key takeaways:
- Always follow the order of operations (PEMDAS).
- Pay close attention to negative signs and exponents.
- Break down complex expressions into smaller, manageable steps.
- Practice consistently to build your skills and confidence.
By incorporating these principles into your problem-solving approach, you'll be well-equipped to excel in mathematics and beyond. Keep learning, keep practicing, and keep exploring the fascinating world of mathematics!