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Answer in paragraph form using complete sentences and include any calculations, graphs, and labeled diagrams as appropriate.

Lizbeth's statement that \((-12)^{-3}\) equals a positive number because the product of two negative integers is a positive integer is incorrect. 

To clarify, let's break down the expression \((-12)^{-3}\). The exponent \(-3\) indicates both the reciprocal and the power. First, the negative exponent means we take the reciprocal of the base, so:

\[
(-12)^{-3} = \frac{1}{(-12)^3}
\]

Now, let's calculate \((-12)^3\):

\[
(-12)^3 = (-12) \times (-12) \times (-12)
\]

The first multiplication, \((-12) \times (-12)\), results in \(144\) because the product of two negative numbers is positive. However, multiplying this result by another \(-12\) yields:

\[
144 \times (-12) = -1728
\]

Thus, \((-12)^3 = -1728\). Therefore, the original expression becomes:

\[
(-12)^{-3} = \frac{1}{-1728} = -\frac{1}{1728}
\]

This final result is a negative number, not a positive one. The reason Lizbeth’s reasoning is incorrect is that while it is true that the product of two negative numbers is positive, an odd number of negative factors will result in a negative product. Since the exponent here is odd (3), the final product is negative. Hence, \((-12)^{-3}\) is negative, not positive.

If you need more details or have any questions, feel free to ask!

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Here are 5 questions related to this explanation:

1. What is the rule for determining the sign of a number raised to an odd exponent?
2. How would the calculation change if the exponent were even instead of odd?
3. Can you find another example where an odd exponent results in a negative number?
4. What is the difference between negative exponents and positive exponents in terms of their impact on the base number?
5. How does the concept of reciprocal apply when dealing with negative exponents?

**Tip:** When dealing with negative exponents, always remember to find the reciprocal of the base and then apply the exponentiation.

Lizbeth says that (-12)^(-3) equals a positive number because the product of two negative integers is a positive integer. Do you agree? Use what you know about exponential expressions to explain.

Explore why (-12)^(-3) equals a negative number, breaking down the steps for calculating the result using properties of exponents and integer multiplication. Suitable for high school students in Grades 9-12.

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