Decoding (-3)^2 - (2 * 5)^2 ÷ 3 A Step-by-Step Guide
In the realm of mathematics, the order of operations is a fundamental concept that dictates the sequence in which mathematical operations should be performed. This ensures consistency and accuracy in calculations, preventing ambiguity and leading to a unique solution for any given expression. The widely recognized acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as a mnemonic to remember this order. In this comprehensive article, we will dissect the expression (-3)^2 - (2 * 5)^2 ÷ 3, meticulously applying the order of operations to arrive at the correct answer. This exploration will not only provide a step-by-step solution but also delve into the underlying principles and potential pitfalls of neglecting the order of operations.
Before we plunge into the specific expression, it's crucial to solidify our understanding of the order of operations. The acronym PEMDAS, or its variations like BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), outlines the hierarchy of operations:
- Parentheses/Brackets: Operations enclosed within parentheses or brackets are performed first. This allows us to group terms and prioritize specific calculations.
- Exponents/Orders: Exponents and roots (also known as orders or indices) are evaluated next. This involves raising a number to a power or finding the root of a number.
- Multiplication and Division: These operations hold equal precedence and are performed from left to right. This means that if both multiplication and division are present, we evaluate them in the order they appear in the expression.
- Addition and Subtraction: Similar to multiplication and division, addition and subtraction have equal precedence and are performed from left to right.
Adhering to this order is paramount in mathematics. Deviating from it can lead to drastically different and incorrect results. The order of operations ensures that mathematical expressions are interpreted consistently, regardless of who is evaluating them.
Now, let's turn our attention to the expression at hand: (-3)^2 - (2 * 5)^2 ÷ 3. We will systematically apply the order of operations to unravel its solution.
Step 1: Parentheses
The first step, according to PEMDAS, is to address operations within parentheses. We have one set of parentheses in the expression: (2 * 5). Performing this multiplication, we get:
(2 * 5) = 10
Our expression now simplifies to:
(-3)^2 - 10^2 ÷ 3
Step 2: Exponents
Next, we tackle exponents. We have two exponential terms: (-3)^2 and 10^2. Let's evaluate them individually.
(-3)^2 = (-3) * (-3) = 9
Note that squaring a negative number results in a positive number. This is because the product of two negative numbers is positive.
10^2 = 10 * 10 = 100
Our expression now transforms into:
9 - 100 ÷ 3
Step 3: Division
Moving on, we encounter a division operation: 100 ÷ 3. Performing this division, we obtain:
100 ÷ 3 = 33.333... (approximately)
For the sake of precision, we will retain the fractional form, 100/3, in our calculations.
Our expression now stands as:
9 - 100/3
Step 4: Subtraction
Finally, we perform the subtraction. To subtract a fraction from a whole number, we need a common denominator. We can rewrite 9 as a fraction with a denominator of 3:
9 = 27/3
Now we can subtract:
27/3 - 100/3 = -73/3
Therefore, the final result of the expression is -73/3, which is approximately equal to -24.333...
One of the most common errors in evaluating mathematical expressions is neglecting the order of operations. For instance, in our example, someone might incorrectly perform the subtraction 9 - 100 before the division 100 ÷ 3, leading to a completely different result. This underscores the critical importance of adhering to PEMDAS or BODMAS.
Another potential pitfall is mishandling negative signs and exponents. It's crucial to remember that (-3)^2 is not the same as -3^2. In the former, the entire -3 is squared, resulting in 9, while in the latter, only the 3 is squared, and the negative sign is applied afterward, resulting in -9.
The order of operations is not merely an arbitrary rule; it's a fundamental principle that underpins the consistency and accuracy of mathematical calculations. Without a standardized order, mathematical expressions would be open to multiple interpretations, leading to confusion and errors. The order of operations ensures that everyone arrives at the same answer when evaluating a given expression.
This consistency is particularly crucial in advanced mathematics, science, and engineering, where complex calculations are commonplace. From solving algebraic equations to modeling physical phenomena, the order of operations is an indispensable tool for ensuring the validity of results.
The order of operations is not confined to the classroom; it has practical applications in various real-world scenarios. For instance, in computer programming, compilers and interpreters rely on the order of operations to correctly evaluate expressions and execute code. In finance, calculating compound interest or investment returns requires a precise understanding of the order of operations.
Even in everyday situations, we implicitly use the order of operations. When calculating the total cost of items with discounts and taxes, we mentally prioritize the discount calculation before adding the tax. This demonstrates the pervasive nature of the order of operations in our daily lives.
In this comprehensive exploration, we have meticulously dissected the expression (-3)^2 - (2 * 5)^2 ÷ 3, demonstrating the critical role of the order of operations in mathematics. By adhering to the PEMDAS/BODMAS principle, we systematically navigated through parentheses, exponents, division, and subtraction to arrive at the correct answer of -73/3. We also highlighted common mistakes, such as neglecting the order of operations or mishandling negative signs and exponents.
The order of operations is not merely a set of rules to memorize; it's a fundamental concept that ensures consistency and accuracy in mathematical calculations. Its importance extends beyond the classroom, permeating various fields, including science, engineering, computer programming, and finance. A solid grasp of the order of operations is essential for anyone seeking to excel in mathematics and related disciplines.
By mastering this principle, we unlock the power to solve complex problems, interpret mathematical expressions with confidence, and communicate mathematical ideas effectively. The order of operations is the bedrock of mathematical consistency, enabling us to build upon fundamental concepts and explore the vast landscape of mathematical knowledge.
