Haley's Work Evaluating ( − 2 ) 8 (-2)^8 ( − 2 ) 8 Is Shown Below: ( − 2 ) 8 = ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) = − 256 \begin{aligned} (-2)^8 & = (-2)(-2)(-2)(-2)(-2)(-2)(-2)(-2) \\ & = -256 \end{aligned} ( − 2 ) 8 ​ = ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) = − 256 ​ 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 fundamental concept that requires a deep understanding of the properties of exponents and the behavior of negative numbers. In this article, we will analyze Haley's work in evaluating the expression $(-2)^8$ and identify her first error.

Haley's Work

Haley's work is shown below:

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

The First Error

At first glance, Haley's work appears to be correct. However, upon closer inspection, we can identify her first error. The error lies in the fact that Haley used the wrong number as a repeated factor. Specifically, she used the number $-2$ as a repeated factor, rather than the number $2$.

Why is this an Error?

When evaluating the expression $(-2)^8$, we need to consider the properties of negative exponents. A negative exponent indicates that we need to take the reciprocal of the base and change the sign of the exponent. In this case, we have $(-2)^8$, which means we need to take the reciprocal of $-2$ and change the sign of the exponent.

However, Haley's work shows that she used the number $-2$ as a repeated factor, rather than the number $2$. This is incorrect because the number $-2$ is not the correct base to use when evaluating the expression $(-2)^8$.

The Correct Approach

To evaluate the expression $(-2)^8$, we need to use the correct base, which is $2$. We can do this by using the property of negative exponents, which states that $(-a)^n = a^n$ for even values of $n$.

Using this property, we can rewrite the expression $(-2)^8$ as $2^8$. We can then evaluate this expression by multiplying $2$ by itself $8$ times:

\begin{aligned} 2^8 & = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \ & = 256 \end{aligned}

Conclusion

In conclusion, Haley's first error was using the wrong number as a repeated factor. Specifically, she used the number $-2$ as a repeated factor, rather than the number $2$. This error led to an incorrect evaluation of the expression $(-2)^8$. By using the correct base and applying the property of negative exponents, we can evaluate the expression correctly and arrive at the correct answer.

Common Mistakes in Evaluating Negative Exponents

When evaluating negative exponents, it is easy to make mistakes. Here are some common mistakes to watch out for:

• Using the wrong base: As we saw in Haley's work, using the wrong base can lead to incorrect evaluations.
• Not applying the property of negative exponents: Failing to apply the property of negative exponents can lead to incorrect evaluations.
• Not considering the sign of the exponent: Failing to consider the sign of the exponent can lead to incorrect evaluations.

Tips for Evaluating Negative Exponents

When evaluating negative exponents, here are some tips to keep in mind:

• Use the correct base: Make sure to use the correct base when evaluating negative exponents.
• Apply the property of negative exponents: Apply the property of negative exponents to evaluate the expression correctly.
• Consider the sign of the exponent: Consider the sign of the exponent when evaluating the expression.

Conclusion

In conclusion, evaluating negative exponents requires a deep understanding of the properties of exponents and the behavior of negative numbers. By using the correct base and applying the property of negative exponents, we can evaluate expressions involving negative exponents correctly. By avoiding common mistakes and following tips for evaluating negative exponents, we can ensure that our evaluations are accurate and reliable.

Final Answer

Introduction

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

Q: What is a negative exponent?

A: A negative exponent is a mathematical expression that involves a base raised to a negative power. For example, $(-2)^{-3}$ is a negative exponent.

Q: How do I evaluate a negative exponent?

A: To evaluate a negative exponent, you need to use the property of negative exponents, which states that $(-a)^n = a^n$ for even values of $n$. You also need to consider the sign of the exponent.

Q: What is the property of negative exponents?

A: The property of negative exponents states that $(-a)^n = a^n$ for even values of $n$. This means that if you have a negative exponent, you can rewrite it as a positive exponent by changing the sign of the base.

Q: How do I apply the property of negative exponents?

A: To apply the property of negative exponents, you need to follow these steps:

1. Check if the exponent is even or odd.
2. If the exponent is even, rewrite the negative exponent as a positive exponent by changing the sign of the base.
3. Evaluate the expression using the positive exponent.

Q: What are some common mistakes to watch out for when evaluating negative exponents?

A: Some common mistakes to watch out for when evaluating negative exponents include:

• Using the wrong base
• Not applying the property of negative exponents
• Not considering the sign of the exponent

Q: How do I avoid these mistakes?

A: To avoid these mistakes, you need to:

• Use the correct base
• Apply the property of negative exponents
• Consider the sign of the exponent

Q: What are some tips for evaluating negative exponents?

A: Some tips for evaluating negative exponents include:

• Use the correct base
• Apply the property of negative exponents
• Consider the sign of the exponent
• Practice, practice, practice!

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

A: Here are some examples of evaluating negative exponents:

• $(-2)^{-3} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
• $(-3)^{-2} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
• $(-4)^{-1} = 4^{-1} = \frac{1}{4}$

Conclusion

In conclusion, evaluating negative exponents requires a deep understanding of the properties of exponents and the behavior of negative numbers. By using the correct base and applying the property of negative exponents, we can evaluate expressions involving negative exponents correctly. By avoiding common mistakes and following tips for evaluating negative exponents, we can ensure that our evaluations are accurate and reliable.

Final Answer

The final answer is: There is no final answer, as this is a Q&A guide. However, the correct answers to the questions are provided above.