Evaluate The Expression 2/3(14.65-4.6) A Step-by-Step Guide

In the realm of mathematics, evaluating expressions is a fundamental skill. This article will serve as a comprehensive guide on how to evaluate the expression ${\frac{2}{3}(14.65-4.6)}$. We will break down the process step-by-step, ensuring a clear understanding of each operation involved. This guide is designed to be beneficial for students, educators, and anyone looking to enhance their mathematical abilities. Understanding the order of operations is crucial in mathematics, and this example provides a practical application of these principles. By the end of this guide, you will have a firm grasp on how to solve similar expressions, boosting your confidence in tackling mathematical problems. Mathematics is more than just numbers and equations; it's a language that describes the world around us. Mastering mathematical skills opens doors to various fields, from science and technology to finance and engineering. So, let's embark on this mathematical journey together and unravel the solution to the given expression.

Understanding the Order of Operations

Before diving into the solution, it's essential to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which mathematical operations should be performed to arrive at the correct answer. Neglecting the order of operations can lead to incorrect results, emphasizing the importance of adhering to this fundamental principle. PEMDAS ensures consistency and accuracy in mathematical calculations, making it a cornerstone of mathematical problem-solving. Think of it as the grammar of mathematics, where the correct order ensures clear and unambiguous communication of mathematical ideas. Each operation has its place in the hierarchy, and following this order is crucial for mathematical fluency. The order of operations is not just a set of rules; it's a logical framework that ensures mathematical expressions are evaluated consistently and accurately.

• Parentheses: Operations inside parentheses are performed first. This step often involves simplifying expressions within the parentheses before moving on to other operations. Parentheses act as containers, prioritizing the operations within them. They help to group terms and clarify the intended order of calculations. Ignoring parentheses can drastically change the outcome of an expression, highlighting their importance in mathematical notation. The presence of parentheses signals a clear priority, guiding the solver to address the enclosed operations before anything else.
• Exponents: Exponents, which indicate repeated multiplication, are evaluated after parentheses. This includes any powers or roots present in the expression. Exponents represent a compact way of expressing repeated multiplication, and their evaluation follows a specific rule within the order of operations. Understanding exponents is crucial for various mathematical concepts, including scientific notation and polynomial expressions. The exponential operation signifies a rapid increase or decrease in value, making it a powerful tool in mathematics.
• Multiplication and Division: These operations are performed from left to right. It's crucial to note that multiplication and division have equal priority, so the order in which they appear in the expression determines the sequence of operations. Treating multiplication and division with equal precedence ensures consistency in calculations. This left-to-right rule avoids ambiguity and maintains the integrity of the mathematical expression. The interplay between multiplication and division is fundamental to arithmetic and algebraic manipulations.
• Addition and Subtraction: Similar to multiplication and division, addition and subtraction are performed from left to right. These operations also have equal priority, and their sequence is determined by their order in the expression. Addition and subtraction are the building blocks of arithmetic, and their correct application is essential for accurate calculations. The left-to-right rule for these operations ensures consistency and prevents misinterpretations. These operations represent the fundamental concepts of combining and separating quantities.

Step-by-Step Solution

Now, let's apply the order of operations to evaluate the expression ${\frac{2}{3}(14.65-4.6)}$. We'll follow the PEMDAS order to ensure accuracy in our calculations. This step-by-step approach will not only solve the expression but also reinforce the application of the order of operations. Each step is crucial in arriving at the correct answer, and understanding the rationale behind each operation is key to mastering mathematical problem-solving. By breaking down the expression into manageable steps, we can gain a deeper understanding of the underlying mathematical principles. This methodical approach is applicable to a wide range of mathematical problems, making it a valuable skill to develop.

Step 1: Simplify the Expression Inside the Parentheses

According to PEMDAS, we begin by simplifying the expression within the parentheses: (14.65 - 4.6). This involves subtracting 4.6 from 14.65. Subtraction is a fundamental arithmetic operation, and performing it within the parentheses is the first step in simplifying the expression. This step isolates the subtraction operation and allows us to deal with it before moving on to other operations. The result of this subtraction will be used in the subsequent steps, highlighting the importance of accuracy in this initial calculation. Parentheses serve as a signal to prioritize the operations within them, ensuring the correct order of evaluation.

14. 65 - 4.6 = 10.05

Therefore, the expression inside the parentheses simplifies to 10.05. This result will now be used in the next step of the evaluation. Reducing the expression inside the parentheses is a crucial step in simplifying the overall expression. It prepares the expression for the next operation, which is multiplication.

Step 2: Multiply by the Fraction

Next, we multiply the result from Step 1 (10.05) by the fraction ${\frac{2}{3}}$. This involves multiplying 10.05 by 2 and then dividing the result by 3. Multiplication and division are key arithmetic operations, and their correct application is essential for solving this expression. This step combines the result from the parentheses with the fractional coefficient, bringing us closer to the final answer. Multiplying by a fraction is equivalent to scaling the value, and understanding this concept is crucial for various mathematical applications. The order of operations dictates that multiplication is performed after simplifying the expression within the parentheses.

${\frac{2}{3} \times 10.05 = \frac{2 \times 10.05}{3}}$

First, multiply 2 by 10.05:

2 * 10.05 = 20.1

Then, divide the result by 3:

20. 1 / 3 = 6.7

Step 3: Final Result

Therefore, the final result of evaluating the expression ${\frac{2}{3}(14.65-4.6)}$ is 6.7. This is the culmination of all the steps, where we have successfully applied the order of operations to simplify the expression. This final answer represents the value of the expression, and it is the result of careful and accurate calculations. Arriving at the final result demonstrates a clear understanding of the mathematical principles involved. This result can be used in further calculations or applications, depending on the context of the problem. The journey from the initial expression to the final result showcases the power of mathematics in solving real-world problems.

Conclusion

In conclusion, evaluating the expression ${\frac{2}{3}(14.65-4.6)}$ demonstrates the importance of understanding and applying the order of operations (PEMDAS). By systematically breaking down the expression into manageable steps, we arrived at the final answer of 6.7. This process not only provides the solution but also reinforces the fundamental principles of mathematics. Mastering these principles is essential for success in various mathematical disciplines. The ability to evaluate expressions accurately is a valuable skill that extends beyond the classroom, impacting problem-solving in everyday life. This example serves as a testament to the power of mathematics in providing clear and concise solutions to complex problems. Mathematics is a journey of discovery, and each expression we solve is a step forward in our understanding of the mathematical world.

This step-by-step guide provides a clear understanding of the process involved in evaluating mathematical expressions. Remember, practice is key to mastering these skills. The more you practice, the more confident you will become in your mathematical abilities. So, keep exploring, keep learning, and keep solving!