Theresa was asked to write an equivalent expression of

Question

that has one positive exponent by applying multiple Laws of Exponents. Her plan to accomplish this is to first apply the Product of a Product Exponent Law, followed by the Product of a Power Exponent Law, and then use the Negative Exponent law to finish.

Use Theresa’s description to rewrite the expression in the order she described.

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Answer 1

To follow Theresa's plan, we can start with a general expression that includes variables with exponents. For the sake of this example, let's use:

\( a^m \times a^n \times (a^p)^q \times a^{-r} \)

1. Apply the Product of a Product Exponent Law: The Product of a Product Exponent Law states that when multiplying like bases, you add the exponents. Additionally, for the power of a power, you multiply the exponents. For the expression, we can group the first two terms together and the third term separately.

   \( a^m \times a^n = a^{m+n} \)

   So now we have:

   \( a^{m+n} \times (a^p)^q \times a^{-r} \)

2. Apply the Product of a Power Exponent Law: The Product of a Power Law states that when raising a power to a power, you multiply the exponents.

   Applying this gives us:

   \( a^{m+n} \times a^{p \cdot q} \times a^{-r} \)

3. Combine all factors using the Product of Powers Law: This means we can now combine all the terms:

   \( a^{(m+n) + (p \cdot q) - r} \)

4. Use the Negative Exponent Law: The Negative Exponent Law states that \( a^{-x} = \frac{1}{a^x} \). If \( (m+n) + (p \cdot q) - r \) is negative, we can express it as:

   \( \frac{1}{a^{-( (m+n) + (p \cdot q) - r )}} \) if necessary. However, we'll keep it as a positive exponent if already positive.

Thus, putting it all together according to Theresa's steps, we arrive at the final expression:

\( a^{(m+n) + (p \cdot q) - r} \)

If you have a specific expression to work with instead of the general form above, please provide that, and I'll apply the steps accordingly.