Evaluating The Expression (0.3)^2 / 2 × 10^5 + (0.3)^2 / 4 × 10^3 × 9.81 - (0.3)^3 / 3 × 10^3 × 9.81 A Step-by-Step Guide

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Evaluating expressions is a fundamental skill in mathematics, forming the bedrock for more advanced concepts. This comprehensive guide will delve into the intricacies of evaluating the expression: (0.3)^2 / 2 × 10^5 + (0.3)^2 / 4 × 10^3 × 9.81 - (0.3)^3 / 3 × 10^3 × 9.81. We'll break down each step, ensuring a clear understanding of the process. Our goal is not just to arrive at the correct answer but to equip you with the knowledge and confidence to tackle similar mathematical challenges. This involves understanding the order of operations, the properties of exponents, and how to manipulate numbers in scientific notation. We will explore each component of the expression in detail, simplifying it progressively to reveal the final solution. Whether you're a student looking to enhance your mathematical abilities or simply someone curious about the world of numbers, this guide offers a detailed and accessible approach to evaluating complex expressions.

Understanding the Expression

Before we begin the evaluation, let's break down the expression into its constituent parts. The expression is:

(0.3)^2 / 2 × 10^5 + (0.3)^2 / 4 × 10^3 × 9.81 - (0.3)^3 / 3 × 10^3 × 9.81

It consists of three main terms, each involving decimal numbers, exponents, multiplication, and division. The presence of 10^5 and 10^3 indicates that we're dealing with numbers in scientific notation, a way of expressing very large or very small numbers concisely. The constant 9.81 is likely related to gravitational acceleration, suggesting a possible physics context. To simplify this expression, we'll follow the order of operations, often remembered by the acronym PEMDAS/BODMAS:

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

This order ensures that we perform the calculations in the correct sequence, leading to the accurate result. Let's begin by tackling the exponents and then proceed with the multiplication and division operations within each term.

Step-by-Step Evaluation

Step 1: Calculate the Exponents

Calculating exponents is the first step in simplifying our expression. We have three exponential terms to evaluate: (0.3)^2 and (0.3)^3. Let's break down each calculation:

  • (0.3)^2 = 0.3 × 0.3 = 0.09
  • (0.3)^3 = 0.3 × 0.3 × 0.3 = 0.027

Now that we have the values of these exponents, we can substitute them back into the original expression. This substitution will make the subsequent calculations more manageable. Replacing (0.3)^2 with 0.09 and (0.3)^3 with 0.027, the expression now looks like this:

0.  09 / 2 × 10^5 + 0.09 / 4 × 10^3 × 9.81 - 0.027 / 3 × 10^3 × 9.81

With the exponents taken care of, we can move on to the next step: performing the divisions within each term. This will further simplify the expression and bring us closer to the final answer.

Step 2: Perform the Divisions

Performing divisions is the next crucial step in evaluating the expression. We have three division operations to handle within the three terms. Let's calculate each one individually:

  • 0.09 / 2 = 0.045
  • 0.09 / 4 = 0.0225
  • 0.027 / 3 = 0.009

Now, substitute these results back into the expression. The expression now becomes:

0.  045 × 10^5 + 0.0225 × 10^3 × 9.81 - 0.009 × 10^3 × 9.81

The expression is now significantly simplified. We've taken care of the exponents and the divisions, and we're left with multiplication and addition/subtraction operations. The next step will be to perform the multiplications, which will further reduce the complexity of the expression.

Step 3: Perform the Multiplications

Performing multiplications is the penultimate step in simplifying the expression. We have three multiplication operations to perform. Let's tackle them one by one:

  • 0.045 × 10^5 = 0.045 × 100000 = 4500
  • 0.0225 × 10^3 × 9.81 = 0.0225 × 1000 × 9.81 = 22.5 × 9.81 = 220.725
  • 0.009 × 10^3 × 9.81 = 0.009 × 1000 × 9.81 = 9 × 9.81 = 88.29

Substituting these results back into the expression, we get:

4500 + 220.725 - 88.29

We have now reduced the expression to a simple addition and subtraction problem. This is the final stage of the evaluation process, and it will lead us to the ultimate answer.

Step 4: Perform the Addition and Subtraction

Performing addition and subtraction is the final step in evaluating our expression. We now have:

4500 + 220.725 - 88.29

Let's perform the addition first:

4500 + 220.725 = 4720.725

Now, subtract 88.29 from the result:

4720.  725 - 88.29 = 4632.435

Therefore, the final result of the expression is 4632.435. We have successfully evaluated the expression by following the order of operations and breaking down the problem into manageable steps.

Final Result

After carefully following each step of the evaluation process, we have arrived at the final result. The value of the expression

(0.3)^2 / 2 × 10^5 + (0.3)^2 / 4 × 10^3 × 9.81 - (0.3)^3 / 3 × 10^3 × 9.81

is:

4632.435

This result is obtained by meticulously applying the order of operations, simplifying exponents, performing divisions and multiplications, and finally, carrying out the addition and subtraction. This comprehensive approach ensures accuracy and provides a clear understanding of the evaluation process.

Conclusion

In conclusion, evaluating mathematical expressions requires a systematic approach and a solid understanding of the order of operations. By breaking down the expression into smaller, manageable steps, we can simplify complex calculations and arrive at the correct answer. In this guide, we meticulously evaluated the expression

(0.3)^2 / 2 × 10^5 + (0.3)^2 / 4 × 10^3 × 9.81 - (0.3)^3 / 3 × 10^3 × 9.81

demonstrating each step in detail. We began by calculating the exponents, followed by divisions, multiplications, and finally, addition and subtraction. This methodical approach not only led us to the correct result of 4632.435 but also provided a clear and concise understanding of the process.

Mastering these fundamental skills is crucial for success in mathematics and related fields. The ability to confidently evaluate expressions forms the foundation for more advanced concepts and problem-solving techniques. By practicing and applying these principles, you can enhance your mathematical abilities and tackle complex challenges with ease.

This guide serves as a testament to the power of breaking down complex problems into simpler steps. Whether you're a student learning the basics or a professional applying mathematical principles, the techniques outlined here will prove invaluable in your journey. Remember, practice and perseverance are key to mastering mathematical skills. Keep exploring, keep learning, and keep pushing your boundaries!