$$ 3^{x+1}=9^{x}.\frac{1}{27} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
3^{x+1} = 9^{x} \cdot \frac{1}{27}
$$

Let's follow the steps below:

### Step 1: Express All Terms with the Same Base

First, notice that all numbers can be expressed as powers of 3:

- $$ 9 = 3^2 $$
- $$ 27 = 3^3 $$

Rewrite the equation using these expressions:

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

### Step 2: Simplify the Exponents

Apply the power rule $$(a^m)^n = a^{mn}$$:

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

Now the equation is:

$$
3^{x+1} = 3^{2x - 3}
$$

### Step 3: Set the Exponents Equal

Since the bases are the same and the equation holds, the exponents must be equal:

$$
x + 1 = 2x - 3
$$

### Step 4: Solve for $$ x $$

Subtract $$ x $$ from both sides:

$$
1 = x - 3
$$

Add 3 to both sides:

$$
x = 4
$$

### Conclusion

The solution to the equation is:

$$
x = 4
$$
$$ x = 4 $$

\( 3^{x+1}=9^{x}.\frac{1}{27} \)