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

Question

UpStudy AI · Accepted Answer

1. Rewrite each term in the numerator so that the exponent has a common base of $$3^{x-2}$$:
   - For $$3^{x+1}$$:
     $$
     3^{x+1} = 3^{(x-2)+3} = 3^{x-2} \cdot 3^3 = 3^{x-2} \cdot 27.
     $$
   - For $$3^{x-1}$$:
     $$
     3^{x-1} = 3^{(x-2)+1} = 3^{x-2} \cdot 3^1 = 3^{x-2} \cdot 3.
     $$

2. Substitute these into the original expression:
   $$
   \frac{3^{x+1} - 4 \cdot 3^{x-1}}{3^{x-2}} = \frac{3^{x-2} \cdot 27 - 4 \cdot (3^{x-2} \cdot 3)}{3^{x-2}}.
   $$

3. Factor out $$3^{x-2}$$ from the numerator:
   $$
   \frac{3^{x-2} \cdot \left(27 - 4 \cdot 3\right)}{3^{x-2}}.
   $$

4. Cancel $$3^{x-2}$$ in the numerator and the denominator:
   $$
   27 - 4 \cdot 3.
   $$

5. Simplify the expression:
   $$
   27 - 12 = 15.
   $$

The simplified result is $$15$$.
The simplified result is $$15$$.

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

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