(1) $$ \frac{\frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}}}{ ext { (3) } \frac{2 \cdot 3^{x-1}+3^{x+1}}{4 \sqrt{9^{x}}}} $$

Question

(1) $$ \frac{\frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}}}{	ext { (3) } \frac{2 \cdot 3^{x-1}+3^{x+1}}{4 \sqrt{9^{x}}}} $$

UpStudy AI · Accepted Answer

Let's simplify the expression step by step:

$$
\frac{\frac{3^{x+1} - 4 \cdot 3^{x-1}}{3^{x-2}}}{\frac{2 \cdot 3^{x-1} + 3^{x+1}}{4 \sqrt{9^{x}}}}
$$

### Step 1: Simplify the Numerator

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

- **Express all terms with the same base:**

$$
3^{x+1} = 3 \cdot 3^x \quad \text{and} \quad 3^{x-1} = \frac{3^x}{3}
$$
$$
3^{x-2} = \frac{3^x}{9}
$$

- **Substitute back into the numerator:**

$$
\frac{3 \cdot 3^x - 4 \cdot \frac{3^x}{3}}{\frac{3^x}{9}} = \frac{3 \cdot 3^x - \frac{4}{3} \cdot 3^x}{\frac{3^x}{9}}
$$

- **Factor out $$3^x$$ in the numerator:**

$$
\frac{3^x \left(3 - \frac{4}{3}\right)}{\frac{3^x}{9}} = \frac{3^x \cdot \frac{5}{3}}{\frac{3^x}{9}}
$$

- **Cancel out $$3^x$$:**

$$
\frac{\frac{5}{3}}{\frac{1}{9}} = \frac{5}{3} \times 9 = 15
$$

### Step 2: Simplify the Denominator

$$
\frac{2 \cdot 3^{x-1} + 3^{x+1}}{4 \sqrt{9^{x}}}
$$

- **Simplify $$\sqrt{9^x}$$:**

$$
\sqrt{9^x} = 9^{\frac{x}{2}} = (3^2)^{\frac{x}{2}} = 3^x
$$

- **Express all terms with the same base:**

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

- **Combine like terms:**

$$
\frac{\frac{2}{3} \cdot 3^x + 3 \cdot 3^x}{4 \cdot 3^x} = \frac{\left(\frac{2}{3} + 3\right) \cdot 3^x}{4 \cdot 3^x} = \frac{\frac{11}{3} \cdot 3^x}{4 \cdot 3^x}
$$

- **Cancel out $$3^x$$:**

$$
\frac{\frac{11}{3}}{4} = \frac{11}{12}
$$

### Step 3: Combine Numerator and Denominator

Now, substitute the simplified numerator and denominator back into the original expression:

$$
\frac{15}{\frac{11}{12}} = 15 \times \frac{12}{11} = \frac{180}{11}
$$

### Final Answer

$$
\frac{180}{11}
$$
The simplified result of the expression is $$ \frac{180}{11} $$.

(1)

Upstudy AI Solution

Answer

Solution

Step 1: Simplify the Numerator

Step 2: Simplify the Denominator

Step 3: Combine Numerator and Denominator

Final Answer

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