Evaluating Algebraic Expressions Step By Step

In mathematics, evaluating algebraic expressions is a fundamental skill. It involves substituting given numerical values for variables within an expression and then performing the necessary arithmetic operations to find the result. This process is crucial in various mathematical contexts, from solving equations to modeling real-world scenarios. In this article, we will delve into the evaluation of algebraic expressions, providing a comprehensive guide with illustrative examples. Let's consider the scenario where we are given the values a = -3, b = -2, and c = 1.5. Our objective is to evaluate several algebraic expressions using these values. We will break down each expression step by step, demonstrating the process of substitution and simplification.

(a) Evaluating a³ - 2b² + 4c

To evaluate the expression a³ - 2b² + 4c, where a = -3, b = -2, and c = 1.5, we will follow a systematic approach, substituting the given values and performing the operations according to the order of precedence (PEMDAS/BODMAS). This involves exponents first, followed by multiplication and division, and finally addition and subtraction. The initial step is to substitute the given values of a, b, and c into the expression. This yields: (-3)³ - 2(-2)² + 4(1.5). The next operation to consider is exponents. Calculate (-3)³ and (-2)². (-3)³ is -3 multiplied by itself three times, resulting in -27. (-2)² is -2 multiplied by itself, which equals 4. Thus, the expression becomes: -27 - 2(4) + 4(1.5). Now, we address the multiplication operations. Multiply 2 by 4, which equals 8, and multiply 4 by 1.5, which equals 6. The expression is now: -27 - 8 + 6. Finally, we perform the addition and subtraction operations from left to right. First, subtract 8 from -27, which results in -35. Then, add 6 to -35, which equals -29. Therefore, the final result of evaluating the expression a³ - 2b² + 4c with the given values is -29. Understanding the order of operations and applying it meticulously ensures accurate evaluation of algebraic expressions. The result of the expression a³ - 2b² + 4c, when a = -3, b = -2, and c = 1.5, is -29. This is achieved by first substituting the given values into the expression, then evaluating the exponents, performing the multiplication, and finally, carrying out the addition and subtraction operations. Each step is crucial in arriving at the correct answer. When dealing with algebraic expressions, it's important to pay close attention to the signs of the numbers, especially when raising negative numbers to powers. A negative number raised to an odd power will result in a negative number, while a negative number raised to an even power will result in a positive number. This principle is clearly illustrated in the calculation of (-3)³ and (-2)² in this example. The ability to accurately evaluate algebraic expressions is a foundational skill in algebra and is essential for solving more complex problems in mathematics and other fields. It requires a solid understanding of the order of operations and careful attention to detail. By breaking down the expression into smaller parts and evaluating each part systematically, one can avoid errors and arrive at the correct solution. This example serves as a clear demonstration of how to apply these principles to evaluate an algebraic expression effectively.

(b) Evaluating (3a - b + 2c)²

To evaluate the expression (3a - b + 2c)², where a = -3, b = -2, and c = 1.5, we again need to follow the order of operations. This means we first simplify the expression inside the parentheses, and then we square the result. The first step is to substitute the values of a, b, and c into the expression. This gives us: (3(-3) - (-2) + 2(1.5))². Next, we perform the multiplications inside the parentheses. 3 multiplied by -3 is -9, and 2 multiplied by 1.5 is 3. The expression inside the parentheses now looks like this: (-9 - (-2) + 3)². Before we proceed further, we need to address the subtraction of a negative number. Subtracting a negative number is the same as adding its positive counterpart, so -(-2) becomes +2. The expression inside the parentheses is now: (-9 + 2 + 3)². Now, we perform the addition and subtraction operations from left to right within the parentheses. -9 plus 2 is -7, and -7 plus 3 is -4. So the expression inside the parentheses simplifies to -4. Now the entire expression looks like this: (-4)². The final step is to square -4, which means multiplying -4 by itself. -4 multiplied by -4 is 16. Therefore, the final result of evaluating the expression (3a - b + 2c)² with the given values is 16. Remember that squaring a negative number results in a positive number because a negative times a negative is a positive. This concept is crucial in understanding how to correctly evaluate expressions involving squares and other even powers. The expression (3a - b + 2c)² is a more complex expression compared to the previous one, as it involves multiple operations within parentheses and then squaring the result. This highlights the importance of following the order of operations meticulously to avoid errors. By breaking the expression down into smaller, more manageable steps, we can ensure that each operation is performed correctly and in the right sequence. This approach is essential for solving more complex algebraic problems. The careful attention to detail required in this type of evaluation is a key skill in mathematics. Understanding how to handle negative signs, fractions, and multiple operations within parentheses is crucial for success in algebra and beyond. The ability to simplify expressions step-by-step allows for a clearer understanding of the problem and reduces the likelihood of making mistakes. Thus, the result of the expression (3a - b + 2c)², when a = -3, b = -2, and c = 1.5, is 16. This was achieved by substituting the given values, simplifying the expression within the parentheses, and then squaring the result. The key to success in evaluating such expressions lies in a thorough understanding of the order of operations and a meticulous approach to each step.

(c) Evaluating (a + 8c) / b

To evaluate the expression (a + 8c) / b, where a = -3, b = -2, and c = 1.5, we again follow the order of operations. In this case, we first simplify the expression in the parentheses, then perform the multiplication, and finally, we divide. The initial step is to substitute the values of a, b, and c into the expression: (-3 + 8(1.5)) / (-2). Next, we perform the multiplication within the parentheses. 8 multiplied by 1.5 is 12. The expression now looks like this: (-3 + 12) / (-2). Now, we perform the addition within the parentheses. -3 plus 12 is 9. The expression simplifies to: 9 / (-2). Finally, we perform the division. 9 divided by -2 is -4.5. Therefore, the final result of evaluating the expression (a + 8c) / b with the given values is -4.5. This example involves a fraction, which adds a layer of complexity to the evaluation process. It's important to remember that division by a negative number results in a negative quotient. Understanding this principle is crucial for arriving at the correct answer. The expression (a + 8c) / b requires us to perform operations both within the parentheses and in the numerator before carrying out the division. This highlights the importance of adhering to the order of operations consistently. By simplifying the expression step-by-step, we can avoid confusion and ensure accuracy. This approach is particularly important when dealing with more complex algebraic expressions that involve multiple operations. The ability to accurately evaluate expressions involving fractions and negative numbers is a key skill in algebra. It requires a solid understanding of the rules of arithmetic and a meticulous approach to each step of the evaluation process. The result of the expression (a + 8c) / b, when a = -3, b = -2, and c = 1.5, is -4.5. This was achieved by substituting the given values, simplifying the expression within the parentheses, performing the multiplication, and then carrying out the division. Each step is essential in reaching the correct solution. When evaluating algebraic expressions, it's crucial to pay close attention to the order of operations and the signs of the numbers involved. A small error in any step can lead to an incorrect result. By breaking down the expression into smaller parts and evaluating each part systematically, we can minimize the risk of making mistakes and ensure that we arrive at the correct answer. Thus, the key to accurately evaluating algebraic expressions lies in a thorough understanding of the principles of arithmetic and a consistent application of the order of operations.

(d) Evaluating (a / b)(4c - 3b)

To evaluate the expression (a / b)(4c - 3b), where a = -3, b = -2, and c = 1.5, we need to follow the order of operations. First, we simplify the expressions within the parentheses, then perform the multiplication. The first step is to substitute the values of a, b, and c into the expression: (-3 / -2)(4(1.5) - 3(-2)). Next, we simplify the expressions within the parentheses. -3 divided by -2 is 1.5. Inside the second set of parentheses, we have 4 multiplied by 1.5, which is 6, and 3 multiplied by -2, which is -6. So the expression becomes: (1.5)(6 - (-6)). Now, we address the subtraction of a negative number within the second set of parentheses. Subtracting a negative number is the same as adding its positive counterpart, so -(-6) becomes +6. The expression now looks like this: (1.5)(6 + 6). We perform the addition within the second set of parentheses. 6 plus 6 is 12. The expression simplifies to: (1.5)(12). Finally, we perform the multiplication. 1.5 multiplied by 12 is 18. Therefore, the final result of evaluating the expression (a / b)(4c - 3b) with the given values is 18. This example involves multiple operations, including division, multiplication, and subtraction, which highlights the importance of following the order of operations meticulously. By breaking the expression down into smaller, more manageable steps, we can ensure that each operation is performed correctly and in the right sequence. This approach is essential for solving more complex algebraic problems. The ability to accurately evaluate expressions involving fractions, negative numbers, and multiple operations is a key skill in algebra. It requires a solid understanding of the rules of arithmetic and a meticulous approach to each step of the evaluation process. The expression (a / b)(4c - 3b) is a good example of how algebraic expressions can combine different operations. The need to perform division, multiplication, and subtraction in the correct order underscores the importance of adhering to the order of operations. A clear understanding of these rules is crucial for avoiding errors and arriving at the correct solution. When evaluating algebraic expressions, it's also important to pay attention to the signs of the numbers involved. A small error in handling negative signs can lead to an incorrect result. By carefully tracking the signs and applying the rules of arithmetic correctly, we can ensure that our calculations are accurate. The result of the expression (a / b)(4c - 3b), when a = -3, b = -2, and c = 1.5, is 18. This was achieved by substituting the given values, simplifying the expressions within the parentheses, performing the division, and then carrying out the multiplication. Each step is essential in reaching the correct solution. Thus, the key to successfully evaluating complex algebraic expressions lies in a thorough understanding of the order of operations, a meticulous approach to each step, and careful attention to detail.

In conclusion, evaluating algebraic expressions requires a systematic approach that involves substituting given values for variables and then performing arithmetic operations in the correct order. By following the order of operations (PEMDAS/BODMAS) and paying close attention to the signs of numbers, we can accurately evaluate even complex expressions. The examples discussed in this article provide a clear illustration of the process and highlight the importance of each step. Mastering this skill is fundamental for success in algebra and beyond.