Simplify (4.4 + 5.2)/6 * (3 - 6)^2 + 7 A Step-by-Step Guide
Introduction
In this article, we will delve into simplifying mathematical expressions, specifically focusing on the expression (4.4 + 5.2)/6 * (3 - 6)^2 + 7. Understanding the order of operations is crucial for accurately solving such problems. We will break down the expression step by step, ensuring clarity and precision in each stage. This guide is designed for students, educators, and anyone looking to enhance their algebraic skills. By the end of this article, you will have a comprehensive understanding of how to simplify this particular expression and similar ones. The key to simplification lies in following the correct sequence of operations, commonly remembered by the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Breaking Down the Expression
To effectively simplify the expression (4.4 + 5.2)/6 * (3 - 6)^2 + 7, it's important to dissect it into manageable parts. The expression comprises several arithmetic operations: addition, subtraction, division, multiplication, and exponentiation. We adhere to the order of operations, often remembered by the acronym PEMDAS/BODMAS. This principle ensures that we perform the calculations in the correct sequence, leading to an accurate result. This involves first addressing the parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Ignoring this order can lead to incorrect answers, which is why understanding and applying PEMDAS/BODMAS is fundamental in mathematical simplification. By following this structured approach, we can systematically reduce the complexity of the expression and arrive at the correct solution. The expression's structure also gives us a roadmap, signaling which parts to prioritize. This careful, step-by-step approach is not only crucial for this specific problem but also for tackling more complex mathematical problems in the future. It's about building a solid foundation in arithmetic and algebraic principles.
Step 1: Simplify within Parentheses
The first step in simplifying this expression is to address the operations within the parentheses. We have two sets of parentheses: (4.4 + 5.2) and (3 - 6). Starting with the first set, we perform the addition: 4.4 + 5.2. This is a straightforward addition of two decimal numbers. Adding these, we get 9.6. Now, let's move to the second set of parentheses, where we have the subtraction: 3 - 6. Subtracting 6 from 3 results in -3. It's crucial to pay attention to the sign here, as it will affect subsequent calculations. After completing these operations, our expression now looks like this: (9.6 / 6) * (-3)^2 + 7. Simplifying within parentheses is a critical step because it organizes the expression, making the remaining operations clearer and more manageable. By dealing with these internal calculations first, we reduce the complexity and pave the way for the next steps in the simplification process. This methodical approach not only ensures accuracy but also develops a good habit of problem-solving in mathematics.
Step 2: Evaluate the Exponent
After simplifying the expressions within the parentheses, the next step in our order of operations is to evaluate the exponent. In our expression, we have (-3)^2, which means -3 raised to the power of 2. This operation involves multiplying -3 by itself: -3 * -3. When multiplying two negative numbers, the result is positive. Therefore, -3 multiplied by -3 equals 9. Now, we replace (-3)^2 with 9 in our expression. Our expression now looks like this: (9.6 / 6) * 9 + 7. Evaluating the exponent is crucial because exponents indicate repeated multiplication, and they must be addressed before multiplication, division, addition, or subtraction. By accurately calculating the exponent, we ensure the correct progression of the simplification process. This step demonstrates the importance of understanding and applying the rules of exponents in mathematical expressions. It's a fundamental concept that underpins many areas of algebra and calculus.
Step 3: Perform Multiplication and Division
With the parentheses and exponent handled, we now turn our attention to multiplication and division. According to the order of operations, these should be performed from left to right. Our expression currently reads (9.6 / 6) * 9 + 7. We first encounter the division operation: 9.6 divided by 6. Performing this division, we get 1.6. Now our expression looks like this: 1.6 * 9 + 7. Next, we perform the multiplication: 1.6 multiplied by 9. This results in 14.4. So, our expression is further simplified to 14.4 + 7. It's important to remember that multiplication and division have equal precedence, so we perform them in the order they appear from left to right. This rule is a cornerstone of the order of operations and ensures consistency in mathematical calculations. By correctly executing the multiplication and division steps, we move closer to the final simplified form of the expression. This stage highlights the importance of precision and attention to detail in arithmetic operations.
Step 4: Perform Addition
Having completed the parentheses, exponent, multiplication, and division operations, the final step in simplifying the expression is to perform the addition. Our expression has been reduced to 14.4 + 7. This is a straightforward addition of a decimal number and a whole number. Adding 14.4 and 7, we get 21.4. Therefore, the simplified value of the original expression (4.4 + 5.2)/6 * (3 - 6)^2 + 7 is 21.4. The addition is the final operation in this particular expression, and it combines the results of all the previous steps. It's crucial to perform addition and subtraction last, according to the order of operations, to ensure the accuracy of the final result. This step underscores the culmination of the simplification process, where all intermediate results are combined to arrive at the solution. The final answer, 21.4, represents the complete simplification of the given mathematical expression.
Conclusion
In conclusion, we have successfully simplified the expression (4.4 + 5.2)/6 * (3 - 6)^2 + 7 by meticulously following the order of operations. We began by simplifying the expressions within the parentheses, then addressed the exponent, followed by multiplication and division, and finally, completed the addition. Each step was crucial in reaching the final simplified value of 21.4. This exercise underscores the importance of adhering to the order of operations in mathematical simplifications. By understanding and applying these principles, we can accurately solve a wide range of algebraic expressions. This step-by-step approach not only helps in achieving the correct answer but also enhances problem-solving skills in mathematics. Mastering the order of operations is fundamental for success in algebra and beyond, and this example provides a clear illustration of how to apply these concepts effectively. The ability to simplify expressions is a valuable skill in various fields, from science and engineering to finance and economics. This article serves as a comprehensive guide to simplifying mathematical expressions and reinforces the significance of precision and methodical problem-solving.