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

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** \( x = 3 \)