Evaluating Mathematical Expressions A Step-by-Step Guide For (3 \cdot(-1))^3+4-1^3

Introduction

In this article, we will delve into the step-by-step evaluation of the mathematical expression $(3 imes (-1))^3 + 4 - 1^3$. This expression combines several fundamental arithmetic operations, including multiplication, exponentiation, addition, and subtraction. Understanding the order of operations is crucial for correctly solving such expressions. We will break down each step, providing a clear explanation to ensure a comprehensive understanding of the evaluation process. This detailed approach will not only help in solving this specific problem but also enhance your ability to tackle similar mathematical expressions with confidence. By meticulously following the order of operations, we can arrive at the correct answer while reinforcing essential mathematical principles.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before diving into the evaluation, it's essential to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS. These acronyms provide a guideline for the sequence in which mathematical operations should be performed:

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

Adhering to this order ensures that mathematical expressions are evaluated consistently and accurately. Failing to follow this order can lead to incorrect results. For example, performing addition before multiplication can drastically change the outcome of an expression. Therefore, a solid grasp of PEMDAS/BODMAS is fundamental to mathematical proficiency. In our given expression, $(3 imes (-1))^3 + 4 - 1^3$, we will meticulously apply these rules to arrive at the correct solution.

Step-by-Step Evaluation

Let's now proceed with the evaluation of the expression $(3 imes (-1))^3 + 4 - 1^3$, meticulously following the order of operations.

Step 1: Parentheses

The first step, according to PEMDAS/BODMAS, is to address the operation within the parentheses:

$3 imes (-1) = -3$

This multiplication results in -3. The expression now becomes:

$(-3)^3 + 4 - 1^3$

Parentheses often group operations that need to be performed before others, and in this case, the multiplication inside the parentheses is a priority. This initial step simplifies the expression and sets the stage for the next operations. By correctly handling the parentheses, we ensure the accuracy of the subsequent steps.

Step 2: Exponents

Next, we address the exponents in the expression. We have two terms with exponents: $(-3)^3$ and $1^3$.

First, let's evaluate $(-3)^3$:

$(-3)^3 = (-3) imes (-3) imes (-3) = -27$

Now, let's evaluate $1^3$:

$1^3 = 1 imes 1 imes 1 = 1$

Substituting these results back into the expression, we get:

$-27 + 4 - 1$

Exponentiation indicates repeated multiplication, and it's crucial to correctly apply the exponent to the base. The evaluation of exponents simplifies the expression further, reducing it to a series of addition and subtraction operations. Understanding how to handle negative numbers raised to powers is particularly important in this step.

Step 3: Addition and Subtraction

Finally, we perform addition and subtraction from left to right. The expression is now:

$-27 + 4 - 1$

First, we add -27 and 4:

$-27 + 4 = -23$

Then, we subtract 1 from -23:

$-23 - 1 = -24$

Therefore, the final result of the expression is -24.

Addition and subtraction are performed sequentially from left to right to ensure the correct outcome. This final step combines the results of the previous operations to arrive at the solution. Paying close attention to the signs of the numbers is essential during this stage.

Conclusion

In conclusion, the evaluation of the expression $(3 imes (-1))^3 + 4 - 1^3$ yields a final result of -24. By meticulously following the order of operations (PEMDAS/BODMAS), we first addressed the parentheses, then the exponents, and finally, the addition and subtraction. Each step was performed with careful attention to detail, ensuring the accuracy of the final result. This methodical approach is crucial for solving mathematical expressions effectively.

Understanding and applying the order of operations is a fundamental skill in mathematics. It not only helps in solving complex expressions but also forms the basis for more advanced mathematical concepts. By mastering these basic principles, one can approach mathematical problems with confidence and precision. The ability to correctly evaluate expressions is a valuable asset in various fields, including science, engineering, and finance. Therefore, practicing and reinforcing these skills is highly beneficial for academic and professional success.

This step-by-step breakdown demonstrates how complex mathematical problems can be simplified by applying a systematic approach. Remember, the key to success in mathematics lies in understanding the fundamental principles and practicing their application. Consistent practice and a clear understanding of the order of operations will undoubtedly lead to improved mathematical proficiency.