Evaluate 4.16 + (-0.387 - (-0.759) - 0.672)^2 Step-by-Step

Introduction

In this article, we will evaluate the expression $4.16 + (-0.387 - (-0.759) - 0.672)^2$. This involves performing arithmetic operations such as addition, subtraction, and exponentiation with decimal numbers. We will follow the order of operations (PEMDAS/BODMAS) to ensure we arrive at the correct result. This comprehensive guide will not only provide the solution but also delve into the step-by-step process, offering a clear understanding of how to tackle such mathematical problems. Whether you're a student looking to enhance your math skills or simply someone who enjoys solving numerical puzzles, this article will provide valuable insights and clarity. Understanding the step-by-step process of evaluating mathematical expressions is crucial for building a strong foundation in arithmetic and algebra. This skill is not only useful in academic settings but also in everyday life where calculations are frequently required. By breaking down the expression into manageable parts and meticulously performing each operation, we can avoid errors and arrive at the correct answer. This article aims to enhance your understanding of how to approach and solve complex arithmetic problems with confidence and accuracy. The following sections will guide you through each step, providing explanations and examples to solidify your understanding.

Step-by-Step Evaluation

Step 1: Simplify the Expression Inside the Parentheses

First, we need to simplify the expression inside the parentheses: $(-0.387 - (-0.759) - 0.672)$. This involves dealing with negative numbers and subtraction. Remember that subtracting a negative number is the same as adding its positive counterpart. Therefore, the expression becomes:

$-0.387 + 0.759 - 0.672$

Now, let's perform the addition and subtraction from left to right:

1. Add -0.387 and 0.759:

$-0.387 + 0.759 = 0.372$

2. Subtract 0.672 from 0.372:

$0.372 - 0.672 = -0.3$

So, the simplified expression inside the parentheses is $-0.3$. This step is crucial as it reduces the complexity of the original expression, making it easier to manage. By carefully handling the signs and performing the operations in the correct order, we ensure the accuracy of our result. The ability to simplify expressions in this manner is a fundamental skill in mathematics, applicable to a wide range of problems. Understanding how to deal with negative numbers and the order of operations is key to mastering arithmetic and algebra. In the next step, we will use this simplified value to continue evaluating the expression.

Step 2: Square the Result from the Parentheses

Next, we need to square the result we obtained from the parentheses, which is $-0.3$. Squaring a number means multiplying it by itself. Therefore, we need to calculate $(-0.3)^2$:

$(-0.3)^2 = (-0.3) imes (-0.3)$

When we multiply two negative numbers, the result is a positive number. So,

$(-0.3) imes (-0.3) = 0.09$

Thus, the square of $-0.3$ is $0.09$. Squaring is an essential operation in mathematics, often used in various contexts such as geometry, algebra, and calculus. It's important to understand that squaring a negative number results in a positive number, which is a key concept in this step. This operation further simplifies the expression, bringing us closer to the final result. By squaring the value obtained from the parentheses, we continue to follow the order of operations, ensuring that the expression is evaluated correctly. The result of this step will be used in the final calculation, where we add it to the remaining term in the expression. Understanding the properties of exponents and how they interact with negative numbers is crucial for mathematical proficiency.

Step 3: Add 4.16 to the Squared Result

Now, we add $4.16$ to the result we obtained from squaring, which is $0.09$. This is the final step in evaluating the expression:

$4.16 + 0.09 = 4.25$

So, the final result of the expression $4.16 + (-0.387 - (-0.759) - 0.672)^2$ is $4.25$. This step combines the results of the previous steps to arrive at the final answer. Addition is a fundamental arithmetic operation, and in this context, it brings together the two main components of the expression: the constant term and the squared result of the parentheses. The ability to perform addition accurately, especially with decimal numbers, is essential for obtaining the correct final result. This final calculation showcases the importance of following the order of operations and performing each step meticulously. By adding the two values, we complete the evaluation of the expression, arriving at a single numerical answer. This result is a decimal number, which is a common outcome in many mathematical expressions involving real numbers. Understanding how to perform addition and other arithmetic operations with precision is crucial for success in mathematics and related fields.

Final Answer

Therefore, $4.16 + (-0.387 - (-0.759) - 0.672)^2 = 4.25$.

Conclusion

In this article, we successfully evaluated the expression $4.16 + (-0.387 - (-0.759) - 0.672)^2$ by following the order of operations. We first simplified the expression inside the parentheses, then squared the result, and finally added it to the remaining term. The final answer is $4.25$. This step-by-step approach demonstrates the importance of breaking down complex problems into smaller, manageable steps. By carefully performing each operation and paying attention to detail, we can confidently arrive at the correct solution. Evaluating mathematical expressions is a fundamental skill that is essential for various applications in science, engineering, and everyday life. Understanding the order of operations and how to handle different types of numbers, such as decimals and negative numbers, is crucial for mathematical proficiency. This article has provided a clear and concise guide to solving this particular expression, but the principles and techniques discussed can be applied to a wide range of mathematical problems. Continuing to practice and refine these skills will lead to greater confidence and competence in mathematics. The ability to evaluate expressions accurately and efficiently is a valuable asset in both academic and professional pursuits.