(2) \( 3 \cdot 3^{x} \) \( 3^{x}+3^{x-1} \)

Certainly! Let's simplify each of the given expressions step by step. --- ### 1. Simplifying \( 3 \cdot 3^{x} \) **Given Expression:** \[ 3 \cdot 3^{x} \] **Simplification Using Exponent Rules:** Recall that when you multiply two powers with the same base, you add their exponents: \[ a^m \cdot a^n = a^{m+n} \] Applying this rule: \[ 3 \cdot 3^{x} = 3^{1} \cdot 3^{x} = 3^{x + 1} \] **Simplified Form:** \[ 3^{x + 1} \] --- ### 2. Simplifying \( 3^{x} + 3^{x-1} \) **Given Expression:** \[ 3^{x} + 3^{x-1} \] **Method 1: Factoring Out the Common Term** 1. **Identify the Common Factor:** Both terms contain \( 3^{x-1} \). Notice that: \[ 3^{x} = 3^{x-1} \cdot 3^{1} = 3^{x-1} \cdot 3 \] 2. **Factor Out \( 3^{x-1} \):** \[ 3^{x} + 3^{x-1} = 3^{x-1} \cdot 3 + 3^{x-1} = 3^{x-1} (3 + 1) \] 3. **Simplify Inside the Parentheses:** \[ 3^{x-1} (3 + 1) = 3^{x-1} \cdot 4 \] **Simplified Form:** \[ 4 \cdot 3^{x-1} \] --- **Alternative Method: Expressing in Terms of \( 3^{x} \)** 1. **Express \( 3^{x-1} \) as \( \frac{3^{x}}{3} \):** \[ 3^{x-1} = \frac{3^{x}}{3} \] 2. **Rewrite the Original Expression:** \[ 3^{x} + 3^{x-1} = 3^{x} + \frac{3^{x}}{3} \] 3. **Factor Out \( 3^{x} \):** \[ 3^{x} \left(1 + \frac{1}{3}\right) = 3^{x} \cdot \frac{4}{3} \] **Simplified Form:** \[ \frac{4}{3} \cdot 3^{x} \] --- **Summary of Simplifications:** 1. \( 3 \cdot 3^{x} = 3^{x + 1} \) 2. \( 3^{x} + 3^{x-1} = 4 \cdot 3^{x-1} \) **or** \( \frac{4}{3} \cdot 3^{x} \) Both forms for the second expression are correct. You can choose the one that best fits the context of your problem or your preference. --- If you have any further questions or need additional clarification, feel free to ask!