Evaluate The Expression $\frac{x^{m+n}}{4m-n}$ Given $x=2$, $y=3$, $m=1$, And $n=2$.

In mathematics, algebraic expressions are fundamental building blocks. These expressions combine variables, constants, and mathematical operations to represent relationships and solve problems. Evaluating an algebraic expression means finding its numerical value by substituting given values for the variables and performing the indicated operations. This article provides a comprehensive guide on how to evaluate algebraic expressions, complete with examples and step-by-step instructions. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will help you master the art of evaluating algebraic expressions.

Understanding the Basics of Algebraic Expressions

Before we dive into evaluating expressions, let's clarify what an algebraic expression is. An algebraic expression is a combination of variables (symbols representing unknown values), constants (fixed numerical values), and mathematical operations (addition, subtraction, multiplication, division, exponents, etc.). For instance, expressions like 3x + 2, y^2 - 4, and (a + b) / c are all algebraic expressions. Each expression consists of terms, which are separated by addition or subtraction. In the expression 3x + 2, 3x and 2 are the terms. Understanding the components of an algebraic expression is crucial for evaluation.

The Role of Variables and Constants

In algebraic expressions, variables are symbols, typically letters (e.g., x, y, a, b), that represent unknown quantities. The value of a variable can change, making it a placeholder for different numbers. On the other hand, constants are fixed numerical values that do not change. For example, in the expression 5x + 7, x is the variable, and 5 and 7 are constants. The coefficient of a variable is the numerical value multiplied by the variable (e.g., in 5x, 5 is the coefficient). Recognizing the role of variables and constants is essential for correctly substituting values during evaluation. Consider the expression 2x^2 - 3x + 1. Here, x is the variable, 2 and 3 are coefficients, and 1 is the constant term. When we evaluate this expression for a specific value of x, we replace x with that value and perform the calculations.

Order of Operations (PEMDAS/BODMAS)

When evaluating algebraic expressions, it's crucial to follow the correct order of operations. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serve as helpful mnemonics. These rules dictate the sequence in which operations should be performed:

1. Parentheses/Brackets: Perform operations inside parentheses or brackets first.
2. Exponents/Orders: Evaluate exponents or orders (powers and square roots).
3. Multiplication and Division: Perform multiplication and division from left to right.
4. Addition and Subtraction: Perform addition and subtraction from left to right.

For example, consider the expression 4 + 3 * (6 - 2)^2. Following PEMDAS, we first evaluate the parentheses: (6 - 2) = 4. Next, we evaluate the exponent: 4^2 = 16. Then, we perform the multiplication: 3 * 16 = 48. Finally, we do the addition: 4 + 48 = 52. Thus, the value of the expression is 52. Mastering the order of operations is vital for accurate evaluation.

Step-by-Step Guide to Evaluating Algebraic Expressions

Evaluating algebraic expressions involves a systematic approach. By following a step-by-step process, you can simplify the expression and arrive at the correct value. Here’s a detailed guide:

Step 1: Substitute the Given Values

The first step in evaluating an algebraic expression is to substitute the given values for the variables. This involves replacing each variable with its corresponding numerical value. For example, if we are given the expression 2x + 3y and the values x = 2 and y = 3, we substitute these values into the expression to get 2(2) + 3(3). It’s crucial to pay close attention to signs (positive and negative) during substitution to avoid errors. Proper substitution sets the stage for the rest of the evaluation process.

Step 2: Apply the Order of Operations

After substituting the values, the next step is to apply the order of operations (PEMDAS/BODMAS) to simplify the expression. This ensures that we perform the operations in the correct sequence. Let's revisit the expression 2(2) + 3(3). According to PEMDAS, we perform the multiplications first: 2(2) = 4 and 3(3) = 9. Now, the expression becomes 4 + 9. Finally, we perform the addition: 4 + 9 = 13. Thus, the value of the expression 2x + 3y when x = 2 and y = 3 is 13. Consistently following the order of operations is key to accurate evaluation.

Step 3: Simplify the Expression

The final step is to simplify the expression to its simplest form. This may involve combining like terms or performing any remaining arithmetic operations. In the previous example, after performing the multiplication and addition, we arrived at the simplified value of 13. However, some expressions may require further simplification. For instance, consider the expression 3(x + 2) - 4x, where x = 5. After substituting, we get 3(5 + 2) - 4(5). Following PEMDAS, we first evaluate the parentheses: (5 + 2) = 7. The expression becomes 3(7) - 4(5). Next, we perform the multiplications: 3(7) = 21 and 4(5) = 20. Now, the expression is 21 - 20. Finally, we perform the subtraction: 21 - 20 = 1. Therefore, the simplified value of the expression is 1. Simplification ensures that the final answer is in its most concise form.

Example Problem: Evaluating $\frac{x^{m+n}}{4m-n}$ when $x=2$, $y=3$, $m=1$, and $n=2$

Let's illustrate the process of evaluating algebraic expressions with a specific example. We will evaluate the expression $\frac{x^{m+n}}{4m-n}$ given that $x=2$, $y=3$, $m=1$, and $n=2$. Note that the value of $y$ is provided but not used in this particular expression. This is a common scenario in mathematical problems, where extra information might be given to test your understanding.

Step 1: Substitute the Given Values

First, we substitute the given values into the expression. We replace $x$ with $2$, $m$ with $1$, and $n$ with $2$. The expression becomes:

$\frac{2^{1+2}}{4(1)-2}$

Step 2: Apply the Order of Operations

Next, we apply the order of operations (PEMDAS/BODMAS) to simplify the expression. We start with the exponent in the numerator:

$1 + 2 = 3$

So, the numerator becomes $2^3$. In the denominator, we perform the multiplication first:

$4(1) = 4$

Now, the expression looks like this:

$\frac{2^3}{4-2}$

We continue by evaluating the exponent:

$2^3 = 2 * 2 * 2 = 8$

And simplify the denominator:

$4 - 2 = 2$

Thus, the expression now is:

$\frac{8}{2}$

Step 3: Simplify the Expression

Finally, we simplify the expression by performing the division:

$\frac{8}{2} = 4$

Therefore, the value of the expression $\frac{x^{m+n}}{4m-n}$ when $x=2$, $m=1$, and $n=2$ is $4$.

This example demonstrates how to systematically evaluate an algebraic expression by substituting values and following the order of operations. By breaking down the problem into manageable steps, we can accurately find the solution.

Common Mistakes to Avoid When Evaluating Expressions

Evaluating algebraic expressions can be challenging, and it’s easy to make mistakes if you’re not careful. Here are some common mistakes to avoid to ensure accuracy:

Forgetting the Order of Operations

One of the most frequent errors is forgetting the order of operations (PEMDAS/BODMAS). Performing operations in the wrong sequence can lead to incorrect results. For example, consider the expression 5 + 3 * 2. If you add 5 and 3 first and then multiply by 2, you’ll get (5 + 3) * 2 = 16, which is wrong. The correct approach is to multiply 3 and 2 first and then add 5, resulting in 5 + (3 * 2) = 11. Always adhere to the order of operations to avoid such mistakes.

Incorrect Substitution

Incorrect substitution of values for variables is another common pitfall. It’s crucial to replace each variable with its corresponding value accurately. For instance, if you have the expression 4x - y and x = 3 and y = -2, substituting incorrectly might lead to 4(3) - 2 = 10, which is wrong. The correct substitution should be 4(3) - (-2) = 12 + 2 = 14. Pay close attention to signs (positive and negative) and double-check your substitutions to prevent errors.

Sign Errors

Sign errors are particularly common when dealing with negative numbers. For example, subtracting a negative number is the same as adding the positive number, and multiplying or dividing by a negative number can change the sign of the result. Consider the expression -3 * (-4). The product of two negative numbers is positive, so the correct answer is 12, not -12. Similarly, be careful when distributing a negative sign across parentheses. For example, -(x - 2) becomes -x + 2, not -x - 2. Vigilance with signs is essential for accurate calculations.

Not Simplifying Completely

Not simplifying completely can also lead to errors or an incomplete answer. Always simplify the expression to its simplest form by combining like terms and performing all possible operations. For example, if you have the expression 2x + 3x - 5, you should combine the like terms 2x and 3x to get 5x - 5. Leaving the expression as 2x + 3x - 5 is not fully simplified. Ensure that you’ve simplified all terms and performed all arithmetic operations to arrive at the most concise answer.

Conclusion

Evaluating algebraic expressions is a fundamental skill in mathematics. By understanding the basics of algebraic expressions, following a step-by-step evaluation process, and avoiding common mistakes, you can master this essential skill. Remember to substitute values carefully, adhere to the order of operations, and simplify expressions completely. With practice, you'll become proficient in evaluating algebraic expressions and build a solid foundation for more advanced mathematical concepts. Whether you're working on homework, preparing for an exam, or simply expanding your mathematical knowledge, the ability to evaluate algebraic expressions accurately is invaluable.