Evaluate A²b + C When A Is -3, B Is 5, And C Is -8
Introduction
In the realm of mathematics, evaluating expressions is a fundamental skill. This article delves into the process of substituting given values into an algebraic expression and simplifying it to obtain a numerical result. Specifically, we will focus on evaluating the expression a²b + c when a = -3, b = 5, and c = -8. This exercise provides a practical application of the order of operations (PEMDAS/BODMAS) and the rules of arithmetic with signed numbers. By understanding how to solve this type of problem, readers can enhance their algebraic manipulation skills and gain confidence in their mathematical abilities.
Understanding the Expression
The expression we aim to evaluate is a²b + c. This expression involves several mathematical operations: exponentiation (a²), multiplication (a² multiplied by b), and addition (adding c to the product of a² and b). To accurately evaluate this expression, we must adhere to the order of operations, which dictates the sequence in which these operations should be performed. The order of operations is typically remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This ensures that we arrive at the correct result by performing the operations in the proper order.
In our expression, a² signifies that we need to square the value of a, which means multiplying it by itself. The term a²b indicates that we need to multiply the result of a² by the value of b. Finally, we add the value of c to the product of a² and b. By carefully following these steps, we can systematically evaluate the expression and determine its numerical value for the given values of a, b, and c. Understanding the structure of the expression and the operations involved is crucial for accurate evaluation.
Step-by-Step Evaluation
Now, let's embark on a detailed, step-by-step evaluation of the expression a²b + c with the given values a = -3, b = 5, and c = -8. This methodical approach will ensure that we correctly apply the order of operations and arrive at the accurate result. Each step will be clearly explained to enhance understanding and provide a solid foundation for similar mathematical problems.
Step 1: Substitute the Given Values
The first crucial step in evaluating any algebraic expression is to substitute the given values for the variables. In our case, we are given a = -3, b = 5, and c = -8. Substituting these values into the expression a²b + c, we get:
(-3)² * 5 + (-8)
This substitution transforms the algebraic expression into a numerical expression, making it ready for the next step in the evaluation process. Accurate substitution is paramount as it sets the stage for the subsequent calculations. A mistake in this initial step can lead to an incorrect final answer. Therefore, it is essential to double-check the substituted values to ensure their accuracy.
Step 2: Evaluate the Exponent
According to the order of operations (PEMDAS/BODMAS), we must address exponents before multiplication and addition. In our expression, the exponent is in the term (-3)², which means we need to square -3. Squaring a number means multiplying it by itself. Therefore,
(-3)² = (-3) * (-3) = 9
It is crucial to remember that when a negative number is squared, the result is positive. This is because multiplying two negative numbers yields a positive number. Substituting this result back into our expression, we now have:
9 * 5 + (-8)
This simplification brings us closer to the final answer by reducing the expression to simpler operations.
Step 3: Perform the Multiplication
Following the order of operations, we now perform the multiplication. In our simplified expression, we have 9 * 5. Multiplying these two numbers together, we get:
9 * 5 = 45
This multiplication step is straightforward, but it is essential to carry it out accurately. Replacing the multiplication term with its result, our expression now becomes:
45 + (-8)
We have now reduced the expression to a simple addition problem, making it easier to find the final answer.
Step 4: Perform the Addition
The final step in evaluating the expression is to perform the addition. We have 45 + (-8). Adding a negative number is the same as subtracting its positive counterpart. Therefore,
45 + (-8) = 45 - 8 = 37
This final calculation gives us the numerical value of the expression for the given values of a, b, and c. By carefully following the order of operations and performing each step accurately, we have successfully evaluated the expression.
Final Answer
After meticulously following the step-by-step evaluation process, we have arrived at the final answer. The value of the expression a²b + c, when a = -3, b = 5, and c = -8, is:
37
This result demonstrates the importance of adhering to the order of operations and performing each calculation with precision. By breaking down the problem into manageable steps and systematically addressing each operation, we have successfully evaluated the algebraic expression.
Conclusion
In conclusion, evaluating the expression a²b + c with a = -3, b = 5, and c = -8 highlights the significance of understanding and applying the order of operations (PEMDAS/BODMAS). By substituting the given values, addressing the exponent, performing the multiplication, and finally, carrying out the addition, we arrived at the solution of 37. This exercise reinforces the fundamental principles of algebraic manipulation and provides a practical example of how to simplify expressions to obtain numerical results. Mastering these skills is essential for success in mathematics and related fields. The ability to accurately evaluate expressions is a valuable tool in problem-solving and critical thinking. This detailed walkthrough serves as a comprehensive guide for readers to enhance their algebraic proficiency and confidence.