Evaluating (x⁵ - X) / (4y) For X = -4 And Y = 4 A Step-by-Step Guide
#mainkeyword Evaluating algebraic expressions is a fundamental skill in mathematics. In this article, we will delve into the process of evaluating the expression (x⁵ - x) / (4y) when x = -4 and y = 4. This involves substituting the given values for the variables and then simplifying the expression using the order of operations. Mastering this skill is crucial for success in algebra and beyond, as it forms the basis for solving equations, inequalities, and more complex mathematical problems. Understanding how to correctly substitute values and simplify expressions ensures accuracy and efficiency in mathematical calculations. We will break down the steps involved in solving this problem, providing a clear and detailed explanation to help you grasp the underlying concepts. This skill is not only essential for academic purposes but also has practical applications in various fields, including engineering, physics, and computer science. By the end of this article, you will have a solid understanding of how to evaluate algebraic expressions with multiple variables and exponents.
Step-by-Step Solution
1. Substitution
The first step in #mainkeyword evaluating the expression (x⁵ - x) / (4y) is to substitute the given values of x and y into the expression. We are given that x = -4 and y = 4. Replacing x and y with these values, we get:
((-4)⁵ - (-4)) / (4 * 4)
This substitution transforms the algebraic expression into a numerical expression, which we can then simplify using the order of operations. The correct substitution is crucial as it sets the stage for the rest of the solution. A mistake in the substitution will lead to an incorrect final answer. The negative sign associated with x = -4 requires careful handling, especially when raising it to the power of 5. Paying close attention to these details ensures that the evaluation proceeds accurately.
2. Simplifying the Numerator
Now, let's focus on simplifying the numerator of the expression, which is (-4)⁵ - (-4). We need to evaluate (-4)⁵ first. Remember that a negative number raised to an odd power results in a negative number. So, (-4)⁵ = -1024.
Therefore, the numerator becomes:
-1024 - (-4)
Subtracting a negative number is the same as adding its positive counterpart:
-1024 + 4 = -1020
The simplification of the numerator involves both exponentiation and addition/subtraction. The exponentiation step, calculating (-4)⁵, is crucial and requires a solid understanding of how exponents work with negative numbers. The subsequent addition of 4 to -1024 is a straightforward arithmetic operation, but it is important to perform it accurately to maintain the correctness of the solution. The result, -1020, is the simplified form of the numerator and will be used in the next step of the evaluation.
3. Simplifying the Denominator
Next, we simplify the denominator of the expression, which is 4 * y. Since y = 4, the denominator becomes:
4 * 4 = 16
The denominator is a simple multiplication operation. Multiplying 4 by 4 gives us 16, which is the simplified form of the denominator. This step is straightforward, but accuracy is still important. The simplified denominator will be used along with the simplified numerator to obtain the final result.
4. Final Calculation
Now that we have simplified both the numerator and the denominator, we can write the expression as:
-1020 / 16
To further simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 4:
(-1020 / 4) / (16 / 4) = -255 / 4
Thus, the final simplified value of the expression is -255/4.
The final calculation involves dividing the simplified numerator by the simplified denominator. The resulting fraction, -1020/16, can be further simplified by dividing both the numerator and the denominator by their greatest common divisor. This step ensures that the final answer is in its simplest form. The division operations must be performed accurately to arrive at the correct final answer. The result, -255/4, is the evaluated value of the given expression for x = -4 and y = 4.
5. Selecting the Correct Option
Comparing our result, -255/4, with the given options:
A. 1023/4 B. 1025/4 C. -1023/4 D. 16385/4
None of the provided options match our calculated result of -255/4. There seems to be an error in the provided options or in the original question's expected answer. Our step-by-step solution clearly leads to -255/4 as the correct evaluation of the expression.
The comparison of the calculated result with the provided options reveals a discrepancy. None of the options match the correct answer of -255/4, indicating a potential error in the options or the expected answer in the original question. This highlights the importance of verifying the final answer and comparing it with the available choices to ensure consistency and accuracy. If such a discrepancy arises in a test or assignment, it is advisable to double-check the calculations and, if necessary, seek clarification from the instructor.
Common Mistakes to Avoid
#mainkeyword Evaluating expressions can be tricky, and students often make mistakes. Here are some common errors to watch out for:
- Incorrect Substitution: Ensure you substitute the values correctly. A simple mistake in substitution can lead to a completely wrong answer.
- Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Exponents should be evaluated before multiplication, division, addition, and subtraction.
- Negative Signs: Pay close attention to negative signs, especially when raising a negative number to a power. For example, (-4)⁵ is negative, but (-4)⁴ is positive.
- Arithmetic Errors: Double-check your arithmetic calculations to avoid simple mistakes in addition, subtraction, multiplication, or division.
- Simplifying Fractions: Make sure to simplify your final fraction to its lowest terms.
Avoiding these common mistakes is crucial for achieving accuracy in evaluating expressions. Each type of mistake can significantly impact the final result, so meticulous attention to detail is necessary throughout the process. Incorrect substitution, for instance, can derail the entire evaluation from the very beginning. Neglecting the order of operations can lead to incorrect intermediate results, which propagate through the rest of the solution. Similarly, mishandling negative signs or committing arithmetic errors can lead to incorrect calculations. Finally, failing to simplify the fraction to its lowest terms may result in an answer that is technically correct but not in its most simplified form. By being aware of these pitfalls and taking steps to avoid them, you can improve your accuracy and confidence in evaluating expressions.
Conclusion
In summary, we evaluated the expression (x⁵ - x) / (4y) when x = -4 and y = 4. By following the steps of substitution, simplifying the numerator and denominator, and performing the final calculation, we arrived at the result -255/4. However, this result does not match any of the provided options, indicating a possible error in the options themselves. #mainkeyword Evaluating algebraic expressions like this requires careful attention to detail and a solid understanding of the order of operations. Remember to double-check your work and watch out for common mistakes to ensure accuracy.
The evaluation process involves a series of steps, each of which requires careful attention to detail. Substitution is the first critical step, where the given values of the variables are plugged into the expression. Simplifying the numerator and denominator involves applying the order of operations correctly, which includes exponentiation, multiplication, and addition/subtraction. The final calculation involves dividing the simplified numerator by the simplified denominator and simplifying the resulting fraction, if necessary. The discrepancy between the calculated result and the provided options underscores the importance of verifying the entire solution process. It also highlights that errors can occur not only in the calculations but also in the problem statement or the answer choices. This comprehensive approach to evaluating expressions ensures that you not only arrive at an answer but also have confidence in its correctness.