Evaluating The Expression 2^8 - 10 - 15 ÷ 3 A Step-by-Step Guide

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  • Introduction
  • Understanding the Order of Operations (PEMDAS/BODMAS)
  • Step-by-Step Evaluation
    • Exponentiation: 2^8
    • Division: 15 ÷ 3
    • Subtraction: 256 - 10
    • Subtraction: 246 - 5
  • Final Result
  • Conclusion

Introduction

In this article, we will delve into the process of evaluating a mathematical expression. The specific expression we will tackle is 2^8 - 10 - 15 ÷ 3. This expression involves a combination of exponentiation, subtraction, and division. To accurately solve this, we must adhere to the order of operations, a fundamental principle in mathematics that dictates the sequence in which operations should be performed. Understanding and applying the correct order of operations is crucial to arriving at the correct answer. The order of operations, often remembered by the acronyms PEMDAS or BODMAS, provides a clear guideline for simplifying expressions. This article will provide a step-by-step breakdown of the evaluation process, ensuring a clear understanding of each operation and its place in the overall calculation. By following this detailed explanation, you will gain a solid grasp of how to approach and solve similar mathematical expressions, enhancing your problem-solving skills in mathematics.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin evaluating the expression, it is essential to understand the order of operations. This is a set of rules that dictate the sequence in which mathematical operations should be performed to ensure consistent and accurate results. The acronyms PEMDAS and BODMAS are commonly used to remember this order. PEMDAS stands for:

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

BODMAS is another acronym that represents the same order of operations:

  • Brackets
  • Orders (exponents)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

The key takeaway here is that operations within parentheses or brackets are performed first, followed by exponents or orders. Then, multiplication and division are carried out from left to right, and finally, addition and subtraction are performed from left to right. Adhering to this order is critical for achieving the correct result in any mathematical expression. In our case, the expression 2^8 - 10 - 15 ÷ 3 involves exponents, subtraction, and division, so we must follow the PEMDAS/BODMAS rule to evaluate it correctly. This understanding forms the foundation for the step-by-step evaluation that follows.

Step-by-Step Evaluation

Now, let's evaluate the expression 2^8 - 10 - 15 ÷ 3 step by step, following the order of operations (PEMDAS/BODMAS). This methodical approach will ensure that we arrive at the correct solution. Each step will be clearly explained to enhance understanding.

Exponentiation: 2^8

The first operation we need to perform is exponentiation. The term 2^8 means 2 raised to the power of 8, which is 2 multiplied by itself 8 times. Let's calculate this:

2^8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

To simplify this, we can break it down:

  • 2 × 2 = 4
  • 4 × 2 = 8
  • 8 × 2 = 16
  • 16 × 2 = 32
  • 32 × 2 = 64
  • 64 × 2 = 128
  • 128 × 2 = 256

So, 2^8 = 256. This result will be used in the next steps of our calculation. Understanding exponentiation is crucial, as it sets the foundation for the rest of the evaluation. The ability to correctly calculate exponents ensures accuracy in the subsequent operations. With the exponentiation complete, we move on to the next operation in the PEMDAS/BODMAS order.

Division: 15 ÷ 3

Following the order of operations, the next operation we need to perform is division. In the expression 2^8 - 10 - 15 ÷ 3, the division operation is 15 ÷ 3. This is a straightforward division, and the result is:

15 ÷ 3 = 5

This step is crucial because it simplifies the expression further, allowing us to proceed with the subtraction operations. Division is a fundamental arithmetic operation, and its correct application is essential for accurate mathematical calculations. In the context of the expression, performing the division before subtraction is dictated by the PEMDAS/BODMAS rule. Now that we have completed the division, we can move on to the subtraction steps. The result of this division, 5, will be used in the subsequent calculations. Each step in the order of operations brings us closer to the final solution, and the correct execution of each operation is vital for achieving an accurate result.

Subtraction: 256 - 10

Now that we have performed the exponentiation and division, we can move on to the subtraction operations. Our expression now looks like this:

256 - 10 - 5

We perform subtraction from left to right. The first subtraction we need to carry out is 256 - 10. This subtraction is relatively simple:

256 - 10 = 246

This step reduces the expression further, making it easier to manage. Subtraction is one of the basic arithmetic operations, and its correct application is crucial for accurate calculations. In the context of the expression, performing the subtraction in the correct order, from left to right, is important. The result of this subtraction, 246, will be used in the final subtraction step. Each step brings us closer to the final solution, and the correct execution of each operation is vital for achieving an accurate result. With this subtraction completed, we have one more step to finalize the evaluation.

Subtraction: 246 - 5

The final operation to perform is the last subtraction. We now have the expression:

246 - 5

Performing this subtraction gives us:

246 - 5 = 241

This completes the evaluation of the original expression. Subtraction, in this final step, leads us to the ultimate result. The accurate execution of this step, following the order of operations, ensures that our final answer is correct. This final subtraction provides the solution to the given mathematical expression. Each step in the process, from exponentiation to division and finally subtraction, has been carefully performed to arrive at this result. The completion of this step marks the end of our evaluation, giving us the final answer.

Final Result

After following the order of operations and performing each step meticulously, we have arrived at the final result. The value of the expression 2^8 - 10 - 15 ÷ 3 is:

241

This is the solution to the expression. The journey from the initial expression to this final result involved several operations, each performed in the correct sequence as dictated by the PEMDAS/BODMAS rule. The final result encapsulates the correct application of these rules and operations. Understanding and adhering to the order of operations is crucial in mathematics, as it ensures accuracy and consistency in calculations. This result serves as a testament to the importance of this principle. The final answer, 241, is the culmination of all the steps and calculations, providing a definitive solution to the problem.

Conclusion

In conclusion, we have successfully evaluated the mathematical expression 2^8 - 10 - 15 ÷ 3 by meticulously following the order of operations (PEMDAS/BODMAS). This process involved exponentiation, division, and subtraction, each performed in the correct sequence to ensure accuracy. The final result of our evaluation is 241. This exercise highlights the importance of adhering to mathematical conventions to arrive at the correct solution. Understanding and applying the order of operations is a fundamental skill in mathematics, essential for solving a wide range of problems. By breaking down the expression into manageable steps and following the established rules, we were able to simplify the expression and determine its value. This step-by-step approach not only provides the correct answer but also enhances comprehension of the underlying mathematical principles. The ability to accurately evaluate mathematical expressions is crucial for various applications in mathematics and other fields. The successful completion of this evaluation reinforces the importance of precision and attention to detail in mathematical calculations.