$$ 1.1 .4 \quad 3 ^ { x + 3 } - 3 ^ { x + 2 } = 486 $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's solve the equation step by step:

$$
3^{x + 3} - 3^{x + 2} = 486
$$

### Step 1: Factor Out the Common Term

Notice that both terms on the left side of the equation have a common factor of $$ 3^{x + 2} $$. We can factor this out:

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

So, the equation becomes:

$$
3^{x + 2} \cdot 2 = 486
$$

### Step 2: Isolate the Exponential Term

Divide both sides of the equation by 2 to isolate $$ 3^{x + 2} $$:

$$
3^{x + 2} = \frac{486}{2} = 243
$$

### Step 3: Express 243 as a Power of 3

We know that:

$$
243 = 3^5
$$

So, the equation now is:

$$
3^{x + 2} = 3^5
$$

### Step 4: Set the Exponents Equal

Since the bases are the same and the equation $$ 3^a = 3^b $$ implies $$ a = b $$, we can set the exponents equal to each other:

$$
x + 2 = 5
$$

### Step 5: Solve for $$ x $$

Subtract 2 from both sides:

$$
x = 5 - 2
$$
$$
x = 3
$$

### **Answer**

$$ x = 3 $$
$$ x = 3 $$

\( 1.1 .4 \quad 3 ^ { x + 3 } - 3 ^ { x + 2 } = 486 \)