Problem 92 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. $$3^{x+2} \cdot 3^{x}=81$$

Step-by-Step Solution

Answer

The solution is \(x = 1\)

Step 1: Rewrite the equation with same base

Use the rule of exponents to simplify \(3^{x+2} \cdot 3^{x}\) as \(3^{x+x+2} = 3^{2x+2}\) and express \(81\) as \(3^4\). The equation becomes \(3^{2x+2} = 3^4\)

Step 2: Equate the powers

Since the bases are equal, their powers must be equal too. Equate the powers to get the equation \(2x+2 = 4\)

Step 3: Solve for x

Subtract 2 from both sides to get \(2x = 2\). Then divide both sides by 2 to find \(x = 1\)