Which Equation Represents The Sentence, one Minus $p$ To The Fourth Power Equals Zero?A) $1-4p=0$ B) \$4p-1=0$[/tex\] C) $1-p^4=0$ D) $p^4-1 \equiv 0$
Understanding the Sentence
The given sentence is "one minus p to the fourth power equals zero." This sentence can be broken down into its individual components to understand the mathematical expression it represents. The key components are "one," "minus," "p to the fourth power," and "equals zero."
Breaking Down the Sentence
- One: This represents the number 1.
- Minus: This is a mathematical operation that represents subtraction.
- p to the fourth power: This represents the exponentiation of p to the power of 4, which can be written as p^4.
- Equals zero: This indicates that the result of the expression should be equal to 0.
Translating the Sentence into a Mathematical Expression
To translate the sentence into a mathematical expression, we need to follow the order of operations (PEMDAS):
- p to the fourth power: This is the first operation, which can be written as p^4.
- Minus: This operation comes next, which can be represented by a minus sign (-).
- One: This is the last component, which can be represented by the number 1.
Writing the Equation
Combining the components, we get the equation: 1 - p^4 = 0.
Evaluating the Options
Now, let's evaluate the given options to determine which one represents the sentence:
A) 1 - 4p = 0 B) 4p - 1 = 0 C) 1 - p^4 = 0 D) p^4 - 1 ≡ 0
Analyzing Option A
Option A is 1 - 4p = 0. This equation is incorrect because it does not represent the sentence "one minus p to the fourth power equals zero." The correct representation of "p to the fourth power" is p^4, not 4p.
Analyzing Option B
Option B is 4p - 1 = 0. This equation is also incorrect because it does not represent the sentence "one minus p to the fourth power equals zero." The correct representation of "p to the fourth power" is p^4, not 4p.
Analyzing Option C
Option C is 1 - p^4 = 0. This equation is correct because it represents the sentence "one minus p to the fourth power equals zero."
Analyzing Option D
Option D is p^4 - 1 ≡ 0. This equation is incorrect because it does not represent the sentence "one minus p to the fourth power equals zero." The correct representation is 1 - p^4 = 0, not p^4 - 1 ≡ 0.
Conclusion
Based on the analysis, the correct equation that represents the sentence "one minus p to the fourth power equals zero" is:
1 - p^4 = 0
This equation is option C.
Frequently Asked Questions
Q: What is the correct representation of "p to the fourth power"?
A: The correct representation of "p to the fourth power" is p^4.
Q: What is the correct equation that represents the sentence "one minus p to the fourth power equals zero"?
A: The correct equation is 1 - p^4 = 0.
Q: Why is option A incorrect?
A: Option A is incorrect because it does not represent the sentence "one minus p to the fourth power equals zero." The correct representation of "p to the fourth power" is p^4, not 4p.
Q: Why is option D incorrect?
A: Option D is incorrect because it does not represent the sentence "one minus p to the fourth power equals zero." The correct representation is 1 - p^4 = 0, not p^4 - 1 ≡ 0.
Final Answer
The final answer is option C: 1 - p^4 = 0.
Understanding the Equation
The equation 1 - p^4 = 0 is a mathematical expression that represents the sentence "one minus p to the fourth power equals zero." This equation is a fundamental concept in algebra and is used to solve various mathematical problems.
Q&A Session
Q: What is the meaning of the equation 1 - p^4 = 0?
A: The equation 1 - p^4 = 0 represents the sentence "one minus p to the fourth power equals zero." This means that the result of subtracting p to the fourth power from 1 is equal to 0.
Q: What is the value of p in the equation 1 - p^4 = 0?
A: The value of p in the equation 1 - p^4 = 0 is not explicitly given. However, we can solve for p by rearranging the equation to isolate p.
Q: How do I solve for p in the equation 1 - p^4 = 0?
A: To solve for p, we can rearrange the equation to isolate p. This can be done by adding p^4 to both sides of the equation, resulting in:
p^4 = 1
Then, we can take the fourth root of both sides to get:
p = ±1
Q: What is the significance of the ± symbol in the equation p = ±1?
A: The ± symbol indicates that p can have two possible values: 1 and -1. This is because the fourth root of 1 can be either 1 or -1.
Q: Can I use the equation 1 - p^4 = 0 to solve for p in a specific problem?
A: Yes, you can use the equation 1 - p^4 = 0 to solve for p in a specific problem. However, you need to ensure that the problem is consistent with the equation and that the solution is valid.
Q: What are some common applications of the equation 1 - p^4 = 0?
A: The equation 1 - p^4 = 0 has various applications in mathematics, science, and engineering. Some common applications include:
- Solving algebraic equations
- Finding roots of polynomials
- Analyzing electrical circuits
- Modeling population growth
Q: Can I use the equation 1 - p^4 = 0 to solve for p in a non-linear problem?
A: Yes, you can use the equation 1 - p^4 = 0 to solve for p in a non-linear problem. However, you need to ensure that the problem is consistent with the equation and that the solution is valid.
Q: What are some common mistakes to avoid when using the equation 1 - p^4 = 0?
A: Some common mistakes to avoid when using the equation 1 - p^4 = 0 include:
- Not checking for extraneous solutions
- Not considering the domain of the equation
- Not using the correct method to solve for p
Conclusion
The equation 1 - p^4 = 0 is a fundamental concept in algebra and has various applications in mathematics, science, and engineering. By understanding the equation and its significance, you can solve for p in a specific problem and apply it to real-world scenarios.
Final Answer
The final answer is that the equation 1 - p^4 = 0 is a fundamental concept in algebra and has various applications in mathematics, science, and engineering.