Select The Correct Answer.Evaluate The Following Expression When $x=-4$ And $y=4$:$\frac{x^5-x}{4}$A. $-\frac{1.029}{4}$B. $\frac{16,385}{4}$C. $\frac{1,025}{4}$D. $\frac{1.097}{4}$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students to master. In this article, we will focus on evaluating the expression x5−x4\frac{x^5-x}{4} when x=−4x=-4 and y=4y=4. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

Understanding the Expression

The given expression is x5−x4\frac{x^5-x}{4}. This expression involves exponentiation, subtraction, and division. To evaluate it, we need to follow the order of operations (PEMDAS):

  1. Parentheses: None in this expression
  2. Exponents: Evaluate the exponentiation
  3. Multiplication and Division: Evaluate from left to right
  4. Addition and Subtraction: Evaluate from left to right

Step 1: Evaluate the Exponentiation

The first step is to evaluate the exponentiation x5x^5. We are given that x=−4x=-4, so we substitute this value into the expression:

(−4)5=−4×−4×−4×−4×−4=−1024(-4)^5 = -4 \times -4 \times -4 \times -4 \times -4 = -1024

Step 2: Subtract x

Next, we subtract xx from the result:

−1024−(−4)=−1024+4=−1020-1024 - (-4) = -1024 + 4 = -1020

Step 3: Divide by 4

Finally, we divide the result by 4:

−10204=−255\frac{-1020}{4} = -255

Conclusion

The final answer is not among the options provided. However, we can simplify the expression to match one of the options. Let's re-evaluate the expression:

x5−x4=−1024−(−4)4=−1024+44=−10204=−255\frac{x^5-x}{4} = \frac{-1024-(-4)}{4} = \frac{-1024+4}{4} = \frac{-1020}{4} = -255

However, we can rewrite the expression as:

x5−x4=−1024+44=−10204=−255=−10204=−255×44=−255\frac{x^5-x}{4} = \frac{-1024+4}{4} = \frac{-1020}{4} = -255 = -\frac{1020}{4} = -\frac{255 \times 4}{4} = -255

But we can also rewrite it as:

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In our previous article, we walked through the process of evaluating the expression $\frac{x^5-x}{4}$ when $x=-4$ and $y=4$. In this article, we will answer some common questions related to evaluating algebraic expressions. **Q: What is the order of operations?** -------------------------------------- A: The order of operations is a set of rules that tells us which operations to perform first when evaluating an expression. The acronym PEMDAS is commonly used to remember the order: 1. **P**arentheses: Evaluate expressions inside parentheses first 2. **E**xponents: Evaluate any exponential expressions next 3. **M**ultiplication and **D**ivision: Evaluate multiplication and division operations from left to right 4. **A**ddition and **S**ubtraction: Finally, evaluate any addition and subtraction operations from left to right **Q: How do I evaluate an expression with exponents?** ------------------------------------------------ A: To evaluate an expression with exponents, you need to follow the order of operations. If the expression has parentheses, evaluate the expression inside the parentheses first. Then, evaluate any exponential expressions next. For example, to evaluate the expression $2^3 \times 4$, you would first evaluate the exponentiation $2^3 = 8$, and then multiply the result by 4. **Q: What is the difference between a variable and a constant?** --------------------------------------------------------- A: A variable is a letter or symbol that represents a value that can change. For example, in the expression $x^2 + 3x$, the variable is $x$. A constant, on the other hand, is a value that does not change. For example, in the expression $x^2 + 3x$, the constant is 3. **Q: How do I simplify an expression?** ----------------------------------------- A: To simplify an expression, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, to simplify the expression $x^2 + 2x + 3x$, you would combine the like terms $2x$ and $3x$ to get $5x$. Then, you would subtract $5x$ from $x^2$ to get $x^2 - 5x$. **Q: What is the difference between an expression and an equation?** --------------------------------------------------------- A: An expression is a group of numbers, variables, and mathematical operations that are combined using addition, subtraction, multiplication, and division. For example, the expression $x^2 + 3x$ is a group of numbers and variables combined using addition and multiplication. An equation, on the other hand, is a statement that says two expressions are equal. For example, the equation $x^2 + 3x = 5$ is a statement that says the expression $x^2 + 3x$ is equal to the expression 5. **Q: How do I evaluate an expression with multiple variables?** --------------------------------------------------------- A: To evaluate an expression with multiple variables, you need to substitute the values of the variables into the expression. For example, to evaluate the expression $x^2 + 3y$, you would substitute the value of $x$ and $y$ into the expression. If $x = 2$ and $y = 3$, then the expression would become $2^2 + 3(3) = 4 + 9 = 13$. **Conclusion** ---------- Evaluating algebraic expressions is a crucial skill for students to master. By following the order of operations and combining like terms, you can simplify complex expressions and solve equations. Remember to substitute the values of variables into the expression and to evaluate expressions with multiple variables carefully. With practice and patience, you will become proficient in evaluating algebraic expressions.