Solving The Expression (-3(7+5))/(3^2) + 9(2) A Step-by-Step Guide
What is the value of the expression (-3(7+5))/(3^2) + 9(2)?
In this article, we will delve into the step-by-step process of evaluating a mathematical expression. Our focus will be on the expression (-3(7+5))/(3^2) + 9(2). We will break down each component, applying the order of operations, to arrive at the final answer. This comprehensive guide aims to provide clarity and understanding, ensuring you can tackle similar mathematical problems with confidence.
Breaking Down the Expression
To effectively solve the expression (-3(7+5))/(3^2) + 9(2), we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures we perform the operations in the correct sequence, leading to an accurate result. Let's begin by dissecting the expression into manageable parts.
Parentheses First
The first step in evaluating the expression is to address the parentheses. Within the parentheses, we have the addition operation (7+5). Performing this addition, we get:
7 + 5 = 12
Now our expression looks like this:
(-3(12))/(3^2) + 9(2)
Addressing the Exponent
Next, we handle the exponent. In our expression, we have 3^2, which means 3 raised to the power of 2. This is equivalent to 3 multiplied by itself:
3^2 = 3 * 3 = 9
Substituting this back into our expression, we have:
(-3(12))/9 + 9(2)
Multiplication and Division
Now we move on to multiplication and division, working from left to right. We have two multiplication operations and one division operation. First, let's address the multiplication in the numerator:
-3 * 12 = -36
Our expression now becomes:
-36/9 + 9(2)
Next, we perform the division:
-36 / 9 = -4
And the remaining multiplication:
9 * 2 = 18
So our expression simplifies to:
-4 + 18
Final Addition
Finally, we perform the addition operation:
-4 + 18 = 14
Therefore, the value of the expression (-3(7+5))/(3^2) + 9(2) is 14. This step-by-step breakdown illustrates the importance of following the order of operations to arrive at the correct solution. Each operation was performed sequentially, ensuring accuracy and clarity in the solution process.
Detailed Step-by-Step Solution
To further solidify our understanding of how to solve the expression (-3(7+5))/(3^2) + 9(2), let's break down each step in meticulous detail. This comprehensive walkthrough will not only provide the solution but also highlight the underlying principles of mathematical operations. By carefully examining each step, we can gain a deeper appreciation for the order of operations and how it governs the outcome of complex expressions.
Step 1: Parentheses
The first step in any mathematical expression following the order of operations (PEMDAS) is to address the parentheses. Inside the parentheses, we have the addition operation (7+5). This is a straightforward addition, and performing it gives us:
7 + 5 = 12
So, the expression now looks like this:
(-3(12))/(3^2) + 9(2)
The parentheses have been simplified, and we can move on to the next operation in the order.
Step 2: Exponents
The next step is to handle any exponents present in the expression. In our case, we have 3^2, which means 3 raised to the power of 2. This is equivalent to multiplying 3 by itself:
3^2 = 3 * 3 = 9
Substituting this value back into the expression, we get:
(-3(12))/9 + 9(2)
The exponent has been evaluated, and we can proceed to multiplication and division.
Step 3: Multiplication and Division (Left to Right)
Now we tackle multiplication and division, performing these operations from left to right. Our expression has two multiplication operations and one division operation. Let's start with the multiplication in the numerator:
-3 * 12 = -36
This simplifies the expression to:
-36/9 + 9(2)
Next, we perform the division:
-36 / 9 = -4
And then the remaining multiplication:
9 * 2 = 18
Now the expression looks much simpler:
-4 + 18
Step 4: Addition
The final step is to perform the addition operation. We have:
-4 + 18 = 14
Therefore, the value of the expression (-3(7+5))/(3^2) + 9(2) is 14. This detailed step-by-step solution underscores the importance of adhering to the order of operations. Each step was carefully executed, ensuring accuracy and clarity in the solution process. By breaking down the expression into smaller, manageable parts, we can confidently solve even complex mathematical problems.
Common Mistakes to Avoid
When evaluating mathematical expressions, particularly those involving multiple operations, it's easy to make mistakes if you're not careful. Understanding common pitfalls can help you avoid them and ensure you arrive at the correct answer. Let's explore some typical errors that occur when solving expressions like (-3(7+5))/(3^2) + 9(2) and how to sidestep them.
Ignoring the Order of Operations
The most prevalent mistake is disregarding the order of operations (PEMDAS/BODMAS). This can lead to performing operations in the wrong sequence, resulting in an incorrect answer. For instance, someone might attempt to add -3 and 7 before addressing the parentheses or the exponent, which is a fundamental error. Remember, parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right) must be followed meticulously.
Incorrectly Handling Parentheses
Parentheses are crucial as they dictate the operations that must be performed first. A common mistake is to overlook the operations within the parentheses or to misinterpret their scope. In our expression, failing to add 7 and 5 before proceeding with other operations would be a significant error. Always simplify the contents of parentheses before moving on.
Errors with Exponents
Exponents can also be a source of mistakes if not handled properly. For example, confusing 3^2 with 3 * 2 would lead to an incorrect result. Remember that an exponent indicates repeated multiplication, so 3^2 means 3 multiplied by itself (3 * 3), not 3 multiplied by 2.
Misunderstanding Multiplication and Division Precedence
Multiplication and division have the same level of precedence and should be performed from left to right. A common error is to perform multiplication before division, even if the division appears earlier in the expression. In our example, dividing -36 by 9 should be done before multiplying 9 by 2 to maintain accuracy.
Sign Errors
Dealing with negative numbers requires careful attention. Sign errors can easily occur if you're not cautious. For example, multiplying a negative number by a positive number results in a negative number, and incorrectly applying this rule can lead to mistakes. In our expression, -3 multiplied by 12 is -36, and any deviation from this can throw off the entire calculation.
Skipping Steps
While it may be tempting to skip steps to save time, this can often lead to errors. Writing out each step clearly helps you keep track of your progress and reduces the likelihood of making mistakes. Especially with complex expressions, taking the time to write out each step can be invaluable.
Not Double-Checking Your Work
Finally, a simple yet effective way to avoid mistakes is to double-check your work. Once you've arrived at an answer, take a moment to review each step to ensure you haven't made any errors. This can help you catch mistakes that you might have otherwise missed.
By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy when evaluating mathematical expressions. Remember to follow the order of operations, handle parentheses and exponents carefully, pay attention to signs, and double-check your work. With practice and attention to detail, you can confidently tackle even the most complex expressions.
Practical Applications of Order of Operations
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is not just a mathematical concept confined to textbooks. It is a fundamental principle that has practical applications in various real-world scenarios. Understanding and applying the order of operations correctly is crucial for accurate calculations and problem-solving in diverse fields. Let's explore some practical applications where this principle plays a vital role.
Financial Calculations
In finance, the order of operations is essential for calculating interest, returns on investments, and various financial metrics. For example, when calculating compound interest, the formula often involves exponents and parentheses. Incorrectly applying the order of operations can lead to significant errors in determining the final amount, which can have serious financial implications. Whether you're calculating loan payments, investment growth, or budgeting expenses, a solid understanding of PEMDAS is crucial for accurate financial planning.
Engineering and Construction
Engineering and construction projects often involve complex calculations that require adherence to the order of operations. Engineers use mathematical models to design structures, calculate loads, and ensure safety. These models frequently involve expressions with parentheses, exponents, and various arithmetic operations. A mistake in the order of operations can lead to structural failures or design flaws, which can have catastrophic consequences. Similarly, in construction, accurate calculations are necessary for material estimation, cost analysis, and project scheduling, all of which rely on the correct application of PEMDAS.
Computer Programming
In computer programming, the order of operations is fundamental for writing code that produces the desired results. Programming languages follow specific rules for evaluating expressions, and these rules are based on the order of operations. When writing code, programmers must ensure that operations are performed in the correct sequence to achieve the intended logic. Failure to do so can result in bugs, incorrect output, or program crashes. From simple calculations to complex algorithms, the order of operations is a cornerstone of programming.
Scientific Research
Scientific research often involves analyzing data and performing statistical calculations. Researchers use mathematical formulas to test hypotheses, interpret results, and draw conclusions. These formulas frequently involve multiple operations, and the order in which they are performed can significantly affect the outcome. For example, in statistical analysis, calculations such as standard deviation, variance, and regression analysis require strict adherence to the order of operations to ensure accurate results. Inaccurate calculations can lead to flawed research findings and incorrect interpretations.
Everyday Problem Solving
Even in everyday situations, the order of operations can be applied to solve practical problems. For example, when calculating the total cost of a purchase with discounts and taxes, it's essential to apply the operations in the correct order. First, calculate the discount, then add the tax, and finally determine the total amount. Similarly, when planning a trip, calculating travel time, distance, and expenses requires a clear understanding of the order of operations. Whether you're managing your personal finances, planning a project, or simply solving a puzzle, the principles of PEMDAS can help you break down complex problems into manageable steps.
In conclusion, the order of operations is a versatile and essential principle that extends far beyond the classroom. Its practical applications span various fields, from finance and engineering to computer programming and everyday problem-solving. By mastering the order of operations, you can enhance your analytical skills, improve your decision-making abilities, and tackle complex challenges with confidence.
Conclusion: Mastering the Expression and Order of Operations
In this comprehensive guide, we have thoroughly explored the process of evaluating the mathematical expression (-3(7+5))/(3^2) + 9(2). We've broken down the expression into manageable parts, meticulously following the order of operations (PEMDAS) to arrive at the solution. By understanding each step—from simplifying parentheses and exponents to performing multiplication, division, addition, and subtraction—we've not only found the answer but also reinforced the fundamental principles of mathematical operations.
Key Takeaways
Throughout our exploration, several key takeaways have emerged. First and foremost, the importance of adhering to the order of operations cannot be overstated. PEMDAS provides a clear framework for solving complex expressions, ensuring that operations are performed in the correct sequence. Neglecting this order can lead to significant errors and incorrect results. Second, a step-by-step approach is crucial for accuracy and clarity. By breaking down the expression into smaller, manageable steps, we can reduce the likelihood of mistakes and gain a deeper understanding of the problem.
Avoiding Common Mistakes
We've also highlighted common mistakes that often occur when evaluating expressions. These include ignoring the order of operations, incorrectly handling parentheses and exponents, misunderstanding multiplication and division precedence, sign errors, skipping steps, and failing to double-check work. By being aware of these pitfalls, you can take proactive measures to avoid them and improve your mathematical proficiency.
Real-World Applications
Moreover, we've discussed the practical applications of the order of operations in various real-world scenarios. From financial calculations and engineering projects to computer programming and scientific research, the principles of PEMDAS are essential for accurate problem-solving and decision-making. Understanding and applying these principles can enhance your analytical skills and enable you to tackle complex challenges with confidence.
Mastering Mathematical Expressions
By mastering the art of evaluating mathematical expressions, you gain a valuable skill that extends far beyond the classroom. Whether you're balancing your budget, designing a structure, writing code, or conducting research, the ability to perform accurate calculations is essential for success. The expression (-3(7+5))/(3^2) + 9(2) serves as an excellent example of how to apply the order of operations and achieve the correct solution.
Final Thoughts
In conclusion, the value of the expression (-3(7+5))/(3^2) + 9(2) is 14. This result is achieved by meticulously following the order of operations and breaking down the expression into manageable steps. By understanding the underlying principles and avoiding common mistakes, you can confidently tackle similar mathematical problems and apply your knowledge to real-world scenarios. Embrace the challenge of mathematical expressions, and you'll find that the skills you develop will serve you well in various aspects of your life.