Study Guide

Learning Objectives

• Rewrite products of powers with a common base using $b^m b^n=b^{m+n}$ (2.4.A.1).
• Explain how $b^{x+k}$ relates to vertical dilation by $b^k$ (2.4.A.1).
• Apply $(b^m)^n=b^{mn}$ and connect $b^{cx}$ to base $(b^c)$ (2.4.A.2).
• Use $b^{-n}=1/b^n$ (2.4.A.3).
• Evaluate $b^{1/k}$ as the $k$th root when it exists (2.4.A.4).

Suggested Practice Sequence

1. Simplify numeric expressions ($2^3\cdot 2^4$, $(3^2)^3$, $5^{-2}$). 2. Rewrite $b^{x+k}$ and $b^{x-k}$ with a factored coefficient. 3. Rewrite $b^{cx}$ as $(b^c)^x$ and compare bases. 4. Convert between radical and rational exponent form. 5. Match symbolic rewrites to horizontal/vertical graph moves.

Classroom / Check-In Moves

• "Same base?" card sort: product vs power vs negative exponent.
• Graph sketch: $2^x$ vs $2^{x+1}$ vs $2^{2x}$.
• Error analysis on illegal exponent addition.

Misconception Watch-List

• Adding exponents across different bases.
• Treating $b^{x+k}$ as $b^x+b^k$.
• $(b^m)^n$ as $b^{m+n}$.
• $b^{-n}$ as $-b^n$.

Vocabulary to Own

• product property; power property; negative exponent
• equivalent form; common base
• horizontal translation; vertical dilation; horizontal dilation
• rational exponent; $k$th root

Materials Map

• Practice: rule drills and rewrites
• Word: interpret rewritten models (weekly vs daily growth)
• Mixed / multi-*: combine 2.3 form with 2.4 manipulation
• Review & practice test: timed simplification fluency
• Interactive: quick rule checks