Solving |17-22|[33 ÷ |6-3| - 2²] A Step-by-Step Guide
Introduction to Order of Operations
In this comprehensive guide, we will delve into the step-by-step solution of the mathematical expression |17-22|[33 ÷ |6-3| - 2²]. Mastering the order of operations is crucial for accurately solving mathematical problems. This expression incorporates several mathematical concepts, including absolute values, division, subtraction, and exponents. To tackle this, we need to follow the PEMDAS/BODMAS rule, which dictates the order in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This structured approach ensures we arrive at the correct answer by addressing each component in the proper sequence. By understanding and applying these rules, we can simplify complex expressions and build a strong foundation in mathematical problem-solving. The importance of adhering to the order of operations cannot be overstated; it is the cornerstone of accurate mathematical calculations and is essential for both academic and practical applications.
Step 1: Solving the Absolute Values
Our first step in simplifying the expression |17-22|[33 ÷ |6-3| - 2²] is to address the absolute values. Absolute value, denoted by vertical bars | |, represents the distance of a number from zero on the number line. It essentially makes any number inside the bars non-negative. Therefore, we need to calculate the values inside the absolute value bars first.
We have two absolute value expressions: |17-22| and |6-3|. Let's start with |17-22|. The difference between 17 and 22 is -5. However, the absolute value of -5, denoted as |-5|, is 5 because absolute value only considers the magnitude or distance from zero, not the direction. Therefore, |17-22| = 5.
Next, we calculate |6-3|. The difference between 6 and 3 is 3. The absolute value of 3, denoted as |3|, is simply 3 since 3 is already a positive number. Thus, |6-3| = 3.
Now, we can substitute these values back into the original expression. The expression becomes 5[33 ÷ 3 - 2²]. This simplification is a crucial step in breaking down the complex expression into manageable parts. By addressing the absolute values first, we've eliminated one layer of complexity and paved the way for the next steps in our calculation. The revised expression is now easier to handle, and we can proceed with the remaining operations following the order of operations.
Step 2: Evaluating the Exponent
Having dealt with the absolute values, our next task in simplifying the expression 5[33 ÷ 3 - 2²] is to evaluate the exponent. According to the order of operations (PEMDAS/BODMAS), exponents come before multiplication, division, addition, and subtraction. In our expression, we have 2², which means 2 raised to the power of 2, or 2 multiplied by itself.
Calculating 2² is straightforward: 2 multiplied by 2 equals 4. Therefore, 2² = 4. Now we can substitute this value back into the expression, replacing 2² with 4. The expression now looks like this: 5[33 ÷ 3 - 4].
By evaluating the exponent, we've further simplified the expression, bringing us closer to the final solution. This step is vital because exponents can significantly impact the outcome of an equation, and addressing them in the correct order is crucial for accurate results. The simplified expression is now more manageable, and we can proceed to the next operations, which include division and subtraction within the brackets, followed by multiplication.
Step 3: Performing the Division
Following the evaluation of the exponent, the next operation to tackle in the expression 5[33 ÷ 3 - 4] is division. According to the order of operations, division and multiplication are performed from left to right. In our expression, we have 33 ÷ 3. This is a straightforward division operation.
When we divide 33 by 3, the result is 11. Therefore, 33 ÷ 3 = 11. Now, we substitute this value back into the expression, replacing 33 ÷ 3 with 11. The expression now becomes 5[11 - 4]. This simplification is another significant step in breaking down the problem into smaller, more manageable parts. By performing the division, we've reduced the complexity within the brackets, paving the way for the final subtraction and multiplication steps.
Division is a fundamental arithmetic operation, and its correct application is crucial in solving mathematical expressions accurately. By addressing the division at this stage, we ensure that we are adhering to the order of operations and progressing towards the correct solution. The updated expression is now simpler, and we can proceed with the subtraction within the brackets, which will further simplify the expression before we perform the final multiplication.
Step 4: Completing the Subtraction
After performing the division, we move on to the subtraction within the brackets in the expression 5[11 - 4]. According to the order of operations, operations within parentheses or brackets are performed before any operations outside of them. In this case, we have the subtraction operation 11 - 4 inside the brackets.
Subtracting 4 from 11 is a basic arithmetic operation. The result of 11 - 4 is 7. Therefore, we can replace the expression inside the brackets with 7. The expression now simplifies to 5[7], which is the same as 5 * 7. This step is crucial because it simplifies the expression significantly, leaving us with a straightforward multiplication problem. By completing the subtraction, we have effectively resolved the operations within the brackets, which is a key step in adhering to the order of operations.
The simplification achieved through this subtraction allows us to move on to the final step, which is the multiplication of 5 by 7. This final operation will give us the ultimate answer to the original expression. By following the order of operations meticulously, we have broken down a complex expression into manageable steps, ensuring accuracy in our calculations and leading us to the correct solution.
Step 5: Performing the Final Multiplication
Having simplified the expression to 5[7], which is equivalent to 5 * 7, we now arrive at the final step: performing the multiplication. According to the order of operations, multiplication is one of the last operations to be carried out, following parentheses, exponents, and division.
Multiplying 5 by 7 is a fundamental arithmetic operation. The result of 5 * 7 is 35. Therefore, the final value of the expression |17-22|[33 ÷ |6-3| - 2²] is 35. This result is the culmination of all the previous steps, each performed in the correct order to ensure accuracy.
By completing the multiplication, we have successfully solved the entire expression. This final step underscores the importance of adhering to the order of operations, as each operation performed in the correct sequence has led us to this conclusive answer. The process of breaking down a complex expression into smaller, manageable steps is a crucial skill in mathematics, and this example demonstrates how to apply it effectively.
Conclusion: The Final Answer
In conclusion, after meticulously following the order of operations (PEMDAS/BODMAS), we have successfully solved the mathematical expression |17-22|[33 ÷ |6-3| - 2²]. Our step-by-step approach involved first addressing the absolute values, then the exponent, followed by division, subtraction, and finally, multiplication. Each step was crucial in simplifying the expression and ensuring an accurate final result.
The final answer to the expression |17-22|[33 ÷ |6-3| - 2²] is 35. This result highlights the importance of adhering to the correct order of operations in mathematics. By breaking down the problem into smaller, manageable parts and addressing each operation in the appropriate sequence, we have demonstrated a clear and effective method for solving complex mathematical expressions.
Understanding and applying the order of operations is a fundamental skill in mathematics. It ensures consistency and accuracy in calculations, making it an essential tool for both academic and practical applications. This detailed walkthrough of the solution not only provides the answer but also serves as a guide for tackling similar mathematical problems in the future. The ability to confidently and accurately solve such expressions is a testament to a strong foundation in mathematical principles.
