Simplify the following expressions (1) $$ \frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{3^{x+1} - 4 \cdot 3^{x-1}}{3^{x-2}}
$$

Let's break it down step by step.

1. **Simplify the Numerator:**

$$
   3^{x+1} - 4 \cdot 3^{x-1}
   $$

- $$3^{x+1} = 3 \cdot 3^{x}$$
   - $$4 \cdot 3^{x-1} = 4 \cdot \frac{3^{x}}{3} = \frac{4}{3} \cdot 3^{x}$$

So, the numerator becomes:

$$
   3 \cdot 3^{x} - \frac{4}{3} \cdot 3^{x} = \left(3 - \frac{4}{3}\right) \cdot 3^{x} = \frac{5}{3} \cdot 3^{x}
   $$

2. **Simplify the Denominator:**

$$
   3^{x-2} = \frac{3^{x}}{3^2} = \frac{3^{x}}{9}
   $$

3. **Combine Numerator and Denominator:**

$$
   \frac{\frac{5}{3} \cdot 3^{x}}{\frac{3^{x}}{9}} = \frac{5}{3} \cdot 3^{x} \cdot \frac{9}{3^{x}} = \frac{5}{3} \cdot 9 = 15
   $$

**Final Simplified Expression:**

$$
15
$$
The simplified expression is $$15$$.

Simplify the following expressions (1) \( \frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}} \)