Evaluating Algebraic Expressions 1/4(c^3 + D^2) For C=-4 And D=10

In this article, we will delve into the process of evaluating algebraic expressions by substituting given values for variables. Specifically, we will focus on the expression 1/4(c^3 + d^2) and determine its value when c = -4 and d = 10. This type of problem is fundamental in algebra and provides a strong foundation for more complex mathematical concepts. By understanding how to correctly substitute and simplify expressions, you will gain a valuable skill applicable across various mathematical domains.

The ability to evaluate expressions is crucial not only in mathematics but also in various real-world applications. From calculating financial investments to determining the trajectory of a projectile, the principles of substitution and simplification are essential tools. This article aims to provide a comprehensive explanation of the steps involved, ensuring clarity and accuracy in your calculations. We will break down the expression into manageable parts, perform the necessary operations, and arrive at the final answer, reinforcing the importance of meticulousness and attention to detail in mathematical problem-solving.

Before we begin the substitution process, it's important to thoroughly understand the expression 1/4(c^3 + d^2). This expression involves several mathematical operations, including exponentiation, addition, and multiplication. The expression essentially states that we need to cube the value of c, square the value of d, add these two results together, and then multiply the sum by 1/4. Each of these operations follows a specific order of precedence, which is crucial to obtaining the correct final answer. For instance, exponentiation must be performed before addition, and the operations within the parentheses must be completed before multiplication.

Breaking down the expression, we can identify the key components: c^3 represents c raised to the power of 3 (or c cubed), and d^2 represents d raised to the power of 2 (or d squared). The values of c and d are given as -4 and 10, respectively. The addition operation combines the results of c^3 and d^2, and finally, the multiplication by 1/4 scales the entire sum down. By dissecting the expression in this manner, we can approach the evaluation systematically and avoid common errors. This step-by-step analysis is a valuable technique in problem-solving, as it allows us to manage complexity by addressing individual components before integrating them into the whole. Understanding the structure of the expression is the first step towards accurate evaluation.

The next critical step in evaluating the expression is the accurate substitution of the given values for the variables. We are given that c = -4 and d = 10. Substituting these values into the expression 1/4(c^3 + d^2), we replace c with -4 and d with 10. This yields the expression 1/4((-4)^3 + (10)^2). This step is foundational, as any error in substitution will propagate through the rest of the calculation, leading to an incorrect result. Therefore, double-checking the substitution is a vital practice to ensure accuracy.

Now, let's analyze the substituted expression. We have 1/4((-4)^3 + (10)^2). The parentheses indicate that the operations inside them must be performed first, following the order of operations (PEMDAS/BODMAS). This means we need to calculate (-4)^3 and (10)^2 before adding them together and then multiplying by 1/4. Careful attention to the signs and exponents is essential at this stage. A negative number raised to an odd power will result in a negative number, while a number raised to the power of 2 is the number multiplied by itself. By meticulously substituting the values and acknowledging the order of operations, we set the stage for the subsequent simplification steps. This process highlights the importance of precision and systematic execution in algebraic evaluations.

Once we have substituted the values, the next crucial step is simplifying the expression according to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our expression, 1/4((-4)^3 + (10)^2), we first need to address the exponents. Let's calculate (-4)^3 and (10)^2 separately. (-4)^3 means -4 multiplied by itself three times: -4 * -4 * -4. -4 * -4 equals 16, and 16 * -4 equals -64. Therefore, (-4)^3 = -64.

Next, we calculate (10)^2, which means 10 multiplied by itself: 10 * 10 = 100. So, (10)^2 = 100. Now, substituting these values back into our expression, we have 1/4(-64 + 100). We now need to perform the addition inside the parentheses: -64 + 100. This is equivalent to 100 - 64, which equals 36. Thus, our expression simplifies to 1/4(36). Finally, we perform the multiplication: 1/4 multiplied by 36. This is the same as dividing 36 by 4, which equals 9. Therefore, the simplified value of the expression is 9. This step-by-step simplification process showcases the methodical approach required to accurately evaluate algebraic expressions.

After diligently substituting the given values and carefully simplifying the expression, we arrive at the final result. We started with the expression 1/4(c^3 + d^2), where c = -4 and d = 10. Through the process of substitution, we transformed the expression into 1/4((-4)^3 + (10)^2). Following the order of operations, we calculated (-4)^3 as -64 and (10)^2 as 100. This led us to the expression 1/4(-64 + 100), which further simplified to 1/4(36). Finally, multiplying 1/4 by 36, we obtained the value 9.

Therefore, the value of the expression 1/4(c^3 + d^2) when c = -4 and d = 10 is 9. This result aligns with option B in the given choices. The journey from the initial expression to the final answer highlights the importance of accuracy, precision, and adherence to mathematical principles. By breaking down the problem into manageable steps and executing each step meticulously, we have successfully evaluated the expression and confirmed the correct answer. This process reinforces the significance of a systematic approach in solving mathematical problems and building confidence in algebraic manipulations.

In conclusion, evaluating algebraic expressions is a fundamental skill in mathematics that involves substituting given values for variables and simplifying the resulting expression according to the order of operations. In this article, we focused on the expression 1/4(c^3 + d^2) and determined its value when c = -4 and d = 10. We meticulously walked through the steps, starting with understanding the expression, substituting the values, simplifying using the order of operations, and arriving at the final result of 9.

This process underscores the importance of accuracy and precision in mathematical calculations. Every step, from the initial substitution to the final simplification, requires careful attention to detail to avoid errors. By breaking down complex expressions into smaller, manageable parts, we can systematically approach problem-solving and increase our confidence in algebraic manipulations. The ability to evaluate expressions is not only crucial in mathematics but also in various fields that require quantitative reasoning. By mastering these skills, we can enhance our problem-solving abilities and apply them to real-world scenarios.

Therefore, the correct answer is B. 9.