> [!summary] This proof uses definitions from the absolute product rule to prove the power rule > **Key Result:** $|x^n|=|x|^n \quad n\in \mathbb{Z^+}$ >[!info]+ Read Time **⏱ 1 min** # Mathematical Proof > [!warning] Assumptions To prove the power rule by [[Mathematical Induction|proof by induction]], assume the following: > - $n$ is a positive [[Integers|integer]] > - The [[Proof of the Absolute Value Product Rule|product rule]] is $|ab| = |a| \cdot|b|$ Prove that $|x^n|=|x|^n \quad n\in \mathbb{Z^+}$ $ \begin{array}{c} \text{Base case n=0}: \\ |x^0| = 1 \quad |x|^0 = 1 \\ \\ \text{Assume it holds for n=k}: \\ |x^k| = |x|^k \\ \\ \text{Show it holds for $n=k+1$}:\\ \begin{align*} |x^{k+1}| &= |x^k| \cdot |x| \\ &\overset{\text{IH}}{=} |x|^k \cdot |x| \\ &\overset{\text{{PR}}}{=} |x|^{k+1} \end{align*} \\\\ \text{Therefore this hold for $n\geq 0$} \end{array} $ --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidian-MathMatter) to support the project and help others discover it! ---