Evaluating (x^6 - X) / (4y) For X = -4 And Y = 4 A Step-by-Step Guide

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This article will guide you through the process of evaluating the expression (x^6 - x) / (4y) when x = -4 and y = 4. We'll break down each step, ensuring a clear understanding of the solution. This is a fundamental concept in algebra, often encountered in various mathematical contexts. Understanding how to substitute values into algebraic expressions is crucial for solving more complex problems in higher mathematics. Let's dive in and learn how to solve this problem effectively.

Understanding the Problem

Before we jump into the solution, let's make sure we fully understand the problem. We are given the algebraic expression:

(x^6 - x) / (4y)

We are also given the values of the variables:

  • x = -4
  • y = 4

Our goal is to substitute these values into the expression and simplify it to find the final answer. This involves following the order of operations (PEMDAS/BODMAS) and being careful with negative signs and exponents. Substitution is a key skill in algebra, allowing us to determine the value of an expression for specific variable values. In this case, we have a rational expression, which means we are dealing with a fraction where the numerator and denominator are both algebraic expressions. Evaluating such expressions requires careful attention to detail and a solid understanding of algebraic manipulation. Mastering these foundational concepts is essential for progressing in mathematics.

Step-by-Step Solution

Now, let's walk through the solution step by step:

Step 1: Substitute the Values

The first step is to substitute the given values of x and y into the expression. Replace x with -4 and y with 4:

((-4)^6 - (-4)) / (4 * 4)

This step is crucial as it sets up the rest of the calculation. Accurate substitution is paramount to obtaining the correct result. A single mistake in this step can lead to a completely wrong answer. We are essentially replacing the variables with their numerical counterparts, allowing us to transition from an algebraic expression to a numerical one.

Step 2: Evaluate the Exponent

Next, we need to evaluate the exponent. (-4)^6 means -4 multiplied by itself six times:

(-4)^6 = (-4) * (-4) * (-4) * (-4) * (-4) * (-4) = 4096

Remember that a negative number raised to an even power results in a positive number. Understanding the rules of exponents is fundamental in algebra. Here, we are dealing with a power of 6, which means we are multiplying -4 by itself six times. This can be a bit tedious to calculate manually, but it's important to understand the underlying principle. In practice, you might use a calculator for such calculations, but it's crucial to grasp the concept of what an exponent represents.

Step 3: Simplify the Numerator

Now we can substitute the value of (-4)^6 back into the expression and simplify the numerator:

(4096 - (-4)) / (4 * 4)

Subtracting a negative number is the same as adding its positive counterpart:

(4096 + 4) / (4 * 4)

4096 + 4 = 4100

So the numerator simplifies to 4100. Simplifying expressions is a core skill in algebra. Here, we are dealing with the subtraction of a negative number, which is a common point of error for many students. Remember the rule: subtracting a negative is the same as adding a positive. This step is crucial for reducing the complexity of the expression and making it easier to work with.

Step 4: Simplify the Denominator

Next, we simplify the denominator:

4 * 4 = 16

The denominator is now 16. This is a straightforward multiplication, but it's important to ensure accuracy in every step. A clear and methodical approach is key to avoiding mistakes in algebraic calculations. We are now closer to the final answer, having simplified both the numerator and the denominator.

Step 5: Divide

Finally, we divide the simplified numerator by the simplified denominator:

4100 / 16 = 1025 / 4

This gives us the final result. Division is the last step in this process, and it's where we combine the simplified numerator and denominator to arrive at the solution. In this case, the result is a fraction, which is a common outcome in algebraic problems. It's often useful to leave the answer as a fraction, especially if it doesn't simplify to a whole number.

Final Answer

Therefore, the value of the expression (x^6 - x) / (4y) when x = -4 and y = 4 is:

1025 / 4

This corresponds to option A. Double-checking your work is always a good practice to ensure you haven't made any mistakes. We have successfully evaluated the expression by following a step-by-step approach, which is a valuable strategy for tackling algebraic problems.

Common Mistakes to Avoid

Evaluating algebraic expressions can be tricky, and there are several common mistakes students often make. Let's highlight some of them to help you avoid them:

  • Incorrect Substitution: As mentioned earlier, accurate substitution is critical. Double-check that you have replaced the variables with the correct values.
  • Sign Errors: Be particularly careful with negative signs. Remember the rules for multiplying and dividing negative numbers. For example, subtracting a negative number is the same as adding a positive number.
  • Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Exponents should be evaluated before multiplication and division, and so on.
  • Exponent Mistakes: Ensure you understand what exponents mean. A common mistake is to multiply the base by the exponent instead of raising the base to the power of the exponent.
  • Simplification Errors: Double-check your simplification steps to avoid arithmetic errors.

By being aware of these common pitfalls, you can significantly improve your accuracy when evaluating algebraic expressions. Practice is key to mastering these skills and developing confidence in your problem-solving abilities.

Practice Problems

To solidify your understanding, try evaluating the following expressions using the same method:

  1. (x^3 + y^2) / (2x) when x = 2 and y = -3
  2. (a^4 - b) / (5b) when a = -1 and b = 5
  3. (m^2 + n^3) / (mn) when m = 3 and n = -2

Working through these practice problems will help you reinforce the concepts and techniques discussed in this article. The more you practice, the more comfortable you will become with evaluating algebraic expressions. Remember to follow the step-by-step approach and pay attention to detail.

Conclusion

Evaluating algebraic expressions is a fundamental skill in mathematics. By understanding the order of operations, paying close attention to signs, and practicing regularly, you can master this skill and build a strong foundation for more advanced mathematical concepts. In this article, we've walked through a detailed solution for evaluating the expression (x^6 - x) / (4y) when x = -4 and y = 4. We've also discussed common mistakes to avoid and provided practice problems to help you further develop your skills. Remember, consistent practice and a methodical approach are the keys to success in algebra. Keep practicing, and you'll become more confident and proficient in evaluating algebraic expressions.