(3) $$ \frac{3^{x+3} \cdot 18^{x}}{6^{x+1} \cdot 9^{x+1}} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{3^{x+3} \cdot 18^{x}}{6^{x+1} \cdot 9^{x+1}}
$$

Let's break it down step by step by expressing all terms in terms of their prime factors.

1. **Express Each Term with Prime Factors:**
   - $$ 18 = 2 \times 3^2 $$
   - $$ 6 = 2 \times 3 $$
   - $$ 9 = 3^2 $$

2. **Rewrite the Numerator and the Denominator:**
   - **Numerator:**
     $$
     3^{x+3} \cdot 18^{x} = 3^{x+3} \cdot (2 \times 3^2)^x = 3^{x+3} \cdot 2^x \cdot 3^{2x} = 2^x \cdot 3^{3x+3}
     $$
   
   - **Denominator:**
     $$
     6^{x+1} \cdot 9^{x+1} = (2 \times 3)^{x+1} \cdot (3^2)^{x+1} = 2^{x+1} \cdot 3^{x+1} \cdot 3^{2x+2} = 2^{x+1} \cdot 3^{3x+3}
     $$

3. **Simplify the Fraction:**
   $$
   \frac{2^x \cdot 3^{3x+3}}{2^{x+1} \cdot 3^{3x+3}} = \frac{2^x}{2^{x+1}} \cdot \frac{3^{3x+3}}{3^{3x+3}} = 2^{-1} \cdot 1 = \frac{1}{2}
   $$

**Final Simplified Form:**

$$
\frac{1}{2}
$$
The simplified form of the expression is $$ \frac{1}{2} $$.

(3)

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