Solving Complex Mathematical Expressions: A Guide to Evaluating [16 - 4 ÷ 2] x 3 - 52 1
Understanding and evaluating complex mathematical expressions is an essential skill for students, researchers, and professionals alike. In this article, we will walk through the process of solving the expression [(16 - 4 ÷ 2) × 3] - 52 using the correct order of operations. This guide will not only help you arrive at the accurate solution but also provide insights into the application of mathematical principles.
Key Concepts: The Order of Operations (PEMDAS/BODMAS)
The order of operations, commonly abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is the set of rules that dictate the sequence in which mathematical operations must be performed. Adhering to these rules ensures that mathematical expressions are evaluated consistently and accurately.
Step-by-Step Solution of the Expression
Let's break down the expression [(16 - 4 ÷ 2) × 3] - 52 into manageable steps:
Evaluate the expression inside the parentheses:16 - 4 ÷ 2 - First, perform the division: 4 ÷ 2 2 - Then, perform the subtraction: 16 - 2 14 - The expression inside the parentheses simplifies to 14. Substitute the simplified value back into the original expression:
(14 × 3) - 52 - Now, perform multiplication: 14 × 3 42 Evaluate the exponent:
52 25 Finally, perform the subtraction:
42 - 25 -7 - Therefore, the value of the expression is -7.
Understanding Each Component
1. Parentheses
The expression inside the parentheses, (16 - 4 ÷ 2), is evaluated first. This is a fundamental aspect of the order of operations. Parentheses are used to group operations that need to be performed before others. In our case, the division within the parentheses is performed before the subtraction.
2. Exponents
The exponent, 52, is evaluated next. In the context of our expression, this is the last operation to be performed as per the order of operations. Once all the values within the parentheses are simplified, we proceed to the exponentiation step.
3. Multiplication and Division (Left to Right)
Multiplication and division are performed from left to right. In our expression, the multiplication step comes after the simplification of the expression within the parentheses. The division 4 ÷ 2 is performed first, and the result is then used in the subtraction step of the primary expression.
4. Addition and Subtraction (Left to Right)
Finally, addition and subtraction are performed from left to right. In our expression, the remaining subtraction step is performed after the multiplication and exponentiation steps have been completed.
Conclusion
Evaluating complex mathematical expressions correctly is crucial for accuracy in various fields, from scientific research to engineering to finance. By following the order of operations (PEMDAS/BODMAS), you can systematically break down expressions into simpler components and arrive at the correct solution. In the case of the expression [(16 - 4 ÷ 2) × 3] - 52, the value is -7.