Calculate \(x^{2} \cdot x^{-5}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)
2.
Calculate \(x^{2n} \cdot x^{3n}\) using the product rule. Explain the logic of your result in complete sentences.
3.
Consider the following presentation:
\begin{equation*}
\begin{aligned}
x^3 \cdot x^{-3} \amp = x^3 \cdot \frac{1}{x^3} \amp \eqnspacer \amp \text{Definition of negative exponents} \\
\amp = \frac{x^3}{x^3} \amp \amp \text{Multiply fractions} \\
\amp = \frac{x \cdot x \cdot x}{x \cdot x \cdot x} \amp \amp \text{Definition of exponents} \\
\amp = \frac{\cancel{x \cdot x \cdot x}}{\cancel{x \cdot x \cdot x}} \amp \amp \text{Reduce the fraction} \\
\amp = 0
\end{aligned}
\end{equation*}
Identify and explain the error. What would you suggest as a way for students to avoid this mistake?
