Haley's Work Evaluating ( − 2 ) 8 (-2)^8 ( − 2 ) 8 Is Shown Below.${ \begin{aligned} (-2)^8 & = (-2)(-2)(-2)(-2)(-2)(-2)(-2)(-2) \ & = -256 \end{aligned} }$Which Statement Best Describes Haley's First Error?A. She Used The Wrong Number As A Repeated

Introduction

In mathematics, evaluating negative exponents is a crucial concept that requires a deep understanding of the underlying principles. Haley's work on evaluating $(-2)^8$ is a great example of how a simple mistake can lead to a significant error. In this article, we will analyze Haley's work and identify her first error.

Haley's Work

Haley's work on evaluating $(-2)^8$ is shown below:

{ \begin{aligned} (-2)^8 & = (-2)(-2)(-2)(-2)(-2)(-2)(-2)(-2) \\ & = -256 \end{aligned} \}

The First Error

Haley's first error is that she used the wrong number as a repeated factor. Specifically, she repeated the number $-2$ eight times, resulting in a product of $-256$. However, this is not the correct way to evaluate $(-2)^8$.

Understanding Negative Exponents

To evaluate $(-2)^8$, we need to understand the concept of negative exponents. A negative exponent is defined as the reciprocal of the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. Therefore, $(-2)^8$ can be rewritten as $\frac{1}{(-2)^8}$.

The Correct Evaluation

Using the definition of negative exponents, we can rewrite $(-2)^8$ as $\frac{1}{(-2)^8}$. Since $(-2)^8 = 256$, we have:

{ \begin{aligned} (-2)^8 & = \frac{1}{(-2)^8} \\ & = \frac{1}{256} \end{aligned} \}

Conclusion

In conclusion, Haley's first error was that she used the wrong number as a repeated factor. She repeated the number $-2$ eight times, resulting in a product of $-256$. However, this is not the correct way to evaluate $(-2)^8$. By understanding the concept of negative exponents, we can rewrite $(-2)^8$ as $\frac{1}{(-2)^8}$ and evaluate it correctly.

Common Mistakes in Evaluating Negative Exponents

There are several common mistakes that students make when evaluating negative exponents. Some of these mistakes include:

• Using the wrong number as a repeated factor: This is the mistake that Haley made in her work. She repeated the number $-2$ eight times, resulting in a product of $-256$.
• Not understanding the concept of negative exponents: Negative exponents are defined as the reciprocal of the positive exponent. Students need to understand this concept in order to evaluate negative exponents correctly.
• Not rewriting the expression correctly: Students need to rewrite the expression with a negative exponent as the reciprocal of the positive exponent.

Tips for Evaluating Negative Exponents

Here are some tips for evaluating negative exponents:

• Understand the concept of negative exponents: Negative exponents are defined as the reciprocal of the positive exponent. Students need to understand this concept in order to evaluate negative exponents correctly.
• Rewrite the expression correctly: Students need to rewrite the expression with a negative exponent as the reciprocal of the positive exponent.
• Use the correct number as a repeated factor: Students need to use the correct number as a repeated factor when evaluating negative exponents.

Conclusion

Introduction

In our previous article, we analyzed Haley's work on evaluating $(-2)^8$ and identified her first error. In this article, we will provide a Q&A guide to help students understand the concept of negative exponents and evaluate them correctly.

Q: What is a negative exponent?

A: A negative exponent is defined as the reciprocal of the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$.

Q: How do I rewrite an expression with a negative exponent?

A: To rewrite an expression with a negative exponent, you need to rewrite it as the reciprocal of the positive exponent. For example, $(-2)^{-8} = \frac{1}{(-2)^8}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: The main difference between a positive exponent and a negative exponent is the direction of the exponent. A positive exponent is raised to the power of the number, while a negative exponent is the reciprocal of the positive exponent.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you need to follow these steps:

1. Rewrite the expression as the reciprocal of the positive exponent.
2. Evaluate the positive exponent.
3. Take the reciprocal of the result.

Q: What are some common mistakes to avoid when evaluating negative exponents?

A: Some common mistakes to avoid when evaluating negative exponents include:

• Using the wrong number as a repeated factor.
• Not understanding the concept of negative exponents.
• Not rewriting the expression correctly.
• Not taking the reciprocal of the positive exponent.

Q: How can I practice evaluating negative exponents?

A: You can practice evaluating negative exponents by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice evaluating negative exponents.

Q: What are some real-world applications of negative exponents?

A: Negative exponents have many real-world applications, including:

• Physics: Negative exponents are used to describe the behavior of particles in quantum mechanics.
• Engineering: Negative exponents are used to describe the behavior of systems in control theory.
• Finance: Negative exponents are used to describe the behavior of investments in finance.

Q: Can you provide some examples of evaluating negative exponents?

A: Here are some examples of evaluating negative exponents:

• $(-2)^{-8} = \frac{1}{(-2)^8} = \frac{1}{256}$
• $(3)^{-4} = \frac{1}{(3)^4} = \frac{1}{81}$
• $(-5)^{-3} = \frac{1}{(-5)^3} = \frac{1}{-125}$

Conclusion

In conclusion, evaluating negative exponents is a crucial concept in mathematics that requires a deep understanding of the underlying principles. By following the steps outlined in this article, you can evaluate negative exponents correctly and avoid common mistakes. Remember to practice evaluating negative exponents regularly to build your skills and confidence.