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

Question

UpStudy AI · Accepted Answer

We begin with the expression

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

**Step 1. Rewrite each power term in the numerator in terms of $$ 3^x $$:**

$$
3^{x+1} = 3^x \cdot 3,
$$

$$
3^{x-1} = \frac{3^x}{3}.
$$

Thus, the numerator becomes

$$
3 \cdot 3^x - 4 \cdot \frac{3^x}{3}.
$$

**Step 2. Factor out $$ 3^x $$ from the numerator:**

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

Combine the coefficients:

$$
3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3}.
$$

So the numerator simplifies to

$$
\frac{5}{3} \cdot 3^x.
$$

**Step 3. Express the denominator in terms of $$ 3^x $$:**

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

**Step 4. Divide the numerator by the denominator:**

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

Cancel $$ 3^x $$ in the numerator and denominator:

$$
\frac{5}{3} \cdot 9.
$$

Multiply the constants:

$$
\frac{5}{3} \cdot 9 = 5 \cdot 3 = 15.
$$

Thus, the expression simplifies to

$$
15.
$$
The expression simplifies to 15.

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