Solving (-4-6)×3-3÷12 A Step-by-Step Math Guide
In mathematics, solving complex expressions requires a systematic approach, following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guide provides a step-by-step solution to the expression (-4-6)×3-3÷12, ensuring clarity and understanding for learners of all levels. Before diving into the specifics of this problem, it's essential to grasp the fundamental principles that govern mathematical calculations. The order of operations is not merely a convention; it's the backbone of mathematical consistency, ensuring that any given expression yields a unique and correct result, regardless of who solves it. This consistency is paramount in fields like engineering, physics, computer science, and finance, where accurate calculations are critical. Parentheses, the P in PEMDAS, take the highest priority. They act as containers, grouping numbers and operations that must be resolved before anything else. This feature is crucial for altering the natural flow of calculation, allowing us to dictate which parts of an expression should be treated as a single unit. Exponents, the E in PEMDAS, represent repeated multiplication and introduce the concept of powers and roots. Handling exponents correctly is essential for dealing with polynomial expressions, scientific notation, and many other areas of mathematics.
Next in line are Multiplication and Division (MD), which share equal priority. These operations are performed from left to right, meaning that whichever comes first in the expression is executed first. This left-to-right rule ensures that expressions like 10 ÷ 2 × 5 are evaluated correctly (5 × 5 = 25) and not as 10 ÷ (2 × 5) (which would incorrectly yield 1). Finally, Addition and Subtraction (AS) also share equal priority and are performed from left to right, similar to multiplication and division. This order ensures consistent evaluation, for example, in expressions involving both positive and negative numbers.
Understanding and adhering to PEMDAS is not just about getting the right answer; it's about developing a structured approach to problem-solving that is applicable across various mathematical contexts. Whether you're balancing a checkbook, calculating the trajectory of a rocket, or designing a bridge, the principles of order of operations are fundamental. Furthermore, a firm grasp of PEMDAS lays the groundwork for more advanced mathematical concepts, including algebra, calculus, and beyond. It is the cornerstone upon which more complex mathematical structures are built.
Step 1: Solving the Parentheses
Our main focus will be on the first part of our problem today, and that is resolving the parentheses. The given expression is (-4-6)×3-3÷12. According to PEMDAS, we must first address the operations within the parentheses. In this case, we have (-4-6), which simplifies to -10. This step is crucial because it groups the negative numbers together, setting the stage for the subsequent multiplication. Misinterpreting or skipping this step would lead to an incorrect result. Parentheses not only dictate the order of operations but also serve to clarify the structure of an expression. They provide visual cues that help in organizing thoughts and preventing errors. For example, in the expression (2 + 3) × 4, the parentheses clearly indicate that the addition must be performed before the multiplication. Without the parentheses, the expression 2 + 3 × 4 would be evaluated differently (3 × 4 first), leading to a different answer.
In more complex expressions, parentheses can be nested within each other, creating layers of operations that must be resolved from the innermost to the outermost. For instance, in the expression 2 × [3 + (4 – 1)], the innermost parentheses (4 – 1) are evaluated first, followed by the addition within the square brackets, and finally the multiplication. Mastering the handling of parentheses is essential for success in algebra, where expressions often involve multiple sets of parentheses and brackets. Moreover, parentheses play a significant role in mathematical notation and the precise representation of mathematical ideas. They are used to define the scope of functions, to group terms in polynomials, and to denote intervals in calculus. In computer programming, parentheses are ubiquitous, used to define the order of operations in arithmetic expressions, to group arguments in function calls, and to control the flow of execution in conditional statements and loops.
Therefore, a thorough understanding of how parentheses work is not just about solving arithmetic problems; it's a fundamental skill that underpins mathematical literacy and proficiency in various STEM fields. It teaches attention to detail, systematic thinking, and the importance of precision in mathematical communication. In the context of problem-solving, parentheses act as a tool for breaking down complex problems into smaller, more manageable parts. By isolating certain operations within parentheses, we can focus on solving them independently, before integrating the results into the larger expression. This divide-and-conquer approach is a powerful strategy in many areas of mathematics and computer science. So, as we tackle mathematical expressions, we should approach them with an appreciation for the role that parentheses play in organizing, clarifying, and ultimately simplifying the process of calculation.
Step 2: Performing Multiplication
Next in our mathematical journey, we have to perform the multiplication. After simplifying the parentheses, our expression now looks like -10 × 3 - 3 ÷ 12. Following PEMDAS, we address multiplication and division from left to right. The first operation we encounter is -10 × 3, which results in -30. This step is a straightforward application of the rules of multiplication with negative numbers. A negative number multiplied by a positive number yields a negative result. Understanding this rule is crucial for handling signed numbers in mathematical calculations. Multiplication, in its essence, is a form of repeated addition. For example, 3 × 4 can be thought of as adding 3 to itself four times (3 + 3 + 3 + 3 = 12). When dealing with negative numbers, this concept extends to subtracting a number repeatedly. For instance, -3 × 4 can be visualized as subtracting 3 from 0 four times (0 – 3 – 3 – 3 – 3 = -12).
In algebra, multiplication is a fundamental operation used in simplifying expressions, solving equations, and working with polynomials. The distributive property, which states that a(b + c) = ab + ac, is a key principle in algebraic manipulation, allowing us to multiply a single term by a group of terms. Multiplication also plays a central role in calculus, particularly in differentiation and integration. The product rule, for example, provides a method for finding the derivative of a function that is the product of two other functions. In geometry, multiplication is used to calculate areas and volumes. The area of a rectangle is found by multiplying its length and width, and the volume of a rectangular prism is calculated by multiplying its length, width, and height. Multiplication is also essential in trigonometry, where trigonometric functions are often expressed as ratios involving multiplication.
In computer science, multiplication is a core arithmetic operation used in algorithms for data processing, image manipulation, and scientific computing. Multiplication is not just a mathematical operation; it's a powerful tool for modeling and solving real-world problems. From calculating compound interest to determining the optimal dosage of a medication, multiplication is a fundamental skill that is applicable across a wide range of disciplines. A strong grasp of multiplication principles is essential for mathematical proficiency and success in STEM fields. As we move forward in our mathematical journey, let's recognize the importance of multiplication as a fundamental building block upon which more complex concepts are built. By mastering the rules of multiplication, we pave the way for understanding advanced mathematical ideas and tackling challenging problems.
Step 3: Performing Division
Continuing our step-by-step guide, we will now perform the division. Following the multiplication, our expression is -30 - 3 ÷ 12. According to PEMDAS, we must perform the division before subtraction. Thus, we divide 3 by 12, which equals 0.25. This step involves understanding the relationship between division and fractions. The expression 3 ÷ 12 can also be written as the fraction 3/12, which simplifies to 1/4, which is equivalent to 0.25 in decimal form. Division is the inverse operation of multiplication. It answers the question of how many times one number is contained within another. For example, 12 ÷ 3 asks how many times 3 fits into 12, which is 4 times. Understanding this relationship is crucial for solving equations and working with ratios and proportions. Division also plays a central role in calculus, particularly in differentiation and integration.
The quotient rule, for example, provides a method for finding the derivative of a function that is the quotient of two other functions. In algebra, division is used to simplify expressions, solve equations, and work with rational expressions. Simplifying fractions, for instance, involves dividing both the numerator and the denominator by their greatest common factor. Division is also essential in statistics, where we calculate measures like the mean (average) by dividing the sum of a set of numbers by the number of values in the set. In computer science, division is a fundamental arithmetic operation used in algorithms for data analysis, signal processing, and scientific computing. The concept of modular division, which finds the remainder of a division operation, is widely used in cryptography and computer programming.
Division is not just a mathematical operation; it's a tool for understanding how quantities relate to each other. From splitting a pizza among friends to calculating the speed of a car, division is a fundamental skill that is applicable in everyday life. A strong grasp of division principles is essential for mathematical proficiency and success in STEM fields. As we continue our mathematical explorations, let's appreciate the power of division as a tool for understanding relationships, solving problems, and making informed decisions. By mastering the rules of division, we equip ourselves with the ability to tackle a wide range of challenges and appreciate the elegance of mathematical reasoning.
Step 4: Performing Subtraction
For the last step in our mathematical journey, we will perform the final subtraction. Our expression is now -30 - 0.25. Subtracting 0.25 from -30 gives us -30.25. This final step combines the results of all previous operations, showcasing the importance of following the order of operations correctly. Subtraction is one of the basic arithmetic operations, representing the removal of objects from a collection. It is the inverse operation of addition, meaning that if we add the amount subtracted back to the result, we should get the original number. For example, if we subtract 5 from 10 to get 5, adding 5 back to 5 gives us 10. Understanding subtraction is crucial for working with negative numbers, algebraic expressions, and solving equations.
In algebra, subtraction is used to simplify expressions, solve equations, and work with polynomials. For example, combining like terms in an algebraic expression involves adding or subtracting their coefficients. Subtraction also plays a role in calculus, particularly in differentiation and integration. The difference rule, for example, provides a method for finding the derivative of a function that is the difference of two other functions. In geometry, subtraction is used to calculate lengths, areas, and volumes. For example, to find the area of a shape formed by removing a smaller shape from a larger one, we subtract the area of the smaller shape from the area of the larger shape.
In statistics, subtraction is used to calculate measures of spread, such as the range (the difference between the highest and lowest values in a dataset). In computer science, subtraction is a fundamental arithmetic operation used in algorithms for data processing, image manipulation, and scientific computing. Subtraction is not just a mathematical operation; it's a tool for comparing quantities, calculating differences, and making informed decisions. From calculating the change from a purchase to determining the distance between two points, subtraction is a fundamental skill that is applicable in everyday life. A strong grasp of subtraction principles is essential for mathematical proficiency and success in STEM fields. As we conclude our mathematical journey, let's recognize the importance of subtraction as a fundamental building block upon which more complex concepts are built. By mastering the rules of subtraction, we pave the way for understanding advanced mathematical ideas and tackling challenging problems.
Final Answer
Therefore, by meticulously following the order of operations (PEMDAS), we have determined that the solution to the expression (-4-6)×3-3÷12 is -30.25. This step-by-step breakdown illustrates the significance of each operation and the critical role of PEMDAS in achieving the correct answer. This comprehensive guide not only provides the solution but also emphasizes the importance of understanding the underlying mathematical principles.