Evaluating |m² + N²| When M = -5 And N = 3 A Step-by-Step Guide

In this article, we will delve into the process of evaluating the expression |m² + n²| given specific values for the variables m and n. This type of problem falls under the domain of basic algebra, where we substitute numerical values into algebraic expressions and simplify to obtain a final result. Specifically, we will consider the case where m = -5 and n = 3. This exercise is crucial for understanding the order of operations, the handling of negative numbers, and the concept of absolute value. The ability to accurately evaluate such expressions forms a cornerstone for more advanced mathematical concepts. We'll break down the process step-by-step, ensuring clarity and providing explanations for each operation. This meticulous approach will allow readers to not only understand the solution to this particular problem but also to generalize the method for similar expressions and values. Let's begin by understanding the core components of the expression.

The expression |m² + n²| involves several key mathematical operations. First, we have the squaring operation (m² and n²), which means multiplying a number by itself. Next, we have the addition operation (+), which combines the results of the squaring operations. Finally, we have the absolute value operation (| |), which gives the magnitude of a number, regardless of its sign. Understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for correctly evaluating expressions. In this case, we first perform the squaring operations, then the addition, and finally take the absolute value. It's also important to pay close attention to the signs of the numbers, especially when dealing with negative numbers. Squaring a negative number results in a positive number, as the product of two negative numbers is positive. This understanding is essential for accurately evaluating the given expression. Let's now substitute the given values of m and n into the expression and proceed with the evaluation. This will provide a concrete example of how to apply the order of operations and handle the signs of numbers within the context of this specific algebraic expression.

Step-by-Step Evaluation

To find the value of the expression |m² + n²| when m = -5 and n = 3, we will substitute these values into the expression and follow the order of operations. This methodical approach ensures that we arrive at the correct answer. First, we replace m with -5 and n with 3 in the expression, resulting in |(-5)² + (3)²|. This substitution is the initial step in transforming the algebraic expression into a numerical one. It's important to ensure that the values are substituted correctly, paying attention to the signs and the positions of the variables. Once the substitution is complete, we can proceed with simplifying the expression by performing the operations in the correct order. The next step involves evaluating the squares of the numbers, which is a fundamental arithmetic operation. Accurate substitution is the cornerstone of evaluating algebraic expressions, as any error in this initial step will propagate through the entire calculation, leading to an incorrect final result. Therefore, careful attention must be paid to this stage to ensure the subsequent steps are based on a solid foundation. Let's move on to the next step, which involves squaring the values we've substituted.

Squaring the Values

Now, we calculate the squares of -5 and 3. Squaring a number means multiplying it by itself. So, (-5)² means -5 multiplied by -5, and (3)² means 3 multiplied by 3. Remember that the product of two negative numbers is positive. Therefore, (-5)² = 25. And (3)² = 9. This step is crucial because it simplifies the expression further, allowing us to proceed with the addition operation. Understanding how to square numbers, especially negative numbers, is fundamental in algebra. A common mistake is to treat (-5)² as -(5)², which would result in -25. However, the parentheses indicate that the entire quantity -5 is being squared. This distinction is vital for accurate calculations. Having squared the values of m and n, we now have |25 + 9|. The expression is becoming simpler, bringing us closer to the final solution. The next step involves adding these two positive numbers together, which is a straightforward arithmetic operation. This step showcases the importance of understanding the rules of arithmetic within the context of algebraic expressions. Let's proceed with the addition.

Adding the Squares

Next, we add the squares: 25 + 9 = 34. So, now we have |34|. This step is a simple addition, but it's a critical step in simplifying the expression. The result of the addition is a positive number, which will be important when we consider the absolute value in the next step. Adding the squares results in a single numerical value inside the absolute value bars. This prepares us for the final operation, which is to determine the absolute value of the result. The addition operation is a fundamental arithmetic skill, and its correct application here is essential for arriving at the correct final answer. It's important to double-check the addition to ensure accuracy, as any error at this stage will affect the final result. Now that we have a single value inside the absolute value bars, we can proceed to the final step, which is finding the absolute value. This will give us the final answer to the problem. Let's proceed with this final step.

Absolute Value

Finally, we take the absolute value of 34. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, the absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. In this case, the absolute value of 34 is simply 34, because 34 is a positive number. Therefore, |34| = 34. This concludes the evaluation of the expression. The absolute value operation ensures that the final result is always non-negative. This is a key property of the absolute value function. Understanding the concept of absolute value is crucial for various mathematical applications, including distance calculations and error analysis. In this particular problem, the absolute value operation provided the final step in simplifying the expression and arriving at the answer. We have now successfully evaluated the expression |m² + n²| for the given values of m and n. Let's summarize the entire process and state the final answer clearly.

Final Answer

Therefore, when m = -5 and n = 3, the value of the expression |m² + n²| is 34. We arrived at this answer by substituting the given values into the expression, squaring the values, adding the squares, and then taking the absolute value of the result. Each step was performed carefully, following the correct order of operations. The understanding of the order of operations, the handling of negative numbers, and the concept of absolute value were crucial in solving this problem. This exercise demonstrates a fundamental concept in algebra, which is the evaluation of expressions given specific values for the variables. The ability to perform such evaluations is essential for more advanced mathematical topics. This problem also reinforces the importance of accuracy in calculations, as a single error in any step can lead to an incorrect final answer. We have presented a step-by-step solution, providing a clear and detailed explanation of each operation. This approach allows readers to understand not only the solution to this particular problem but also the general method for evaluating similar expressions. In conclusion, the value of the expression |m² + n²| when m = -5 and n = 3 is 34.