Problem 94 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \(x=1\)

Step 1: Simplify the Equation

We know that in mathematics, we can add exponents when we multiply terms with the same base. Therefore, \(3^{x+2} \cdot 3^{x}\) simplifies to \(3^{2x+2}\). Rewriting our equation, we get: \(3^{2x+2}=81\)

Step 2: Write 81 with a base of 3

It is important to have the same base on both sides of the equation to compare exponents. Thus, we rewrite 81 as \(3^4\). Thus, the equation becomes: \(3^{2x+2}=3^4\)

Step 3: Use exponentiation rules

With same bases on both sides, we arrive at: \(2x+2=4\)

Step 4: Subtract 2 from both sides

Subtracting 2 from both sides, results in: \(2x=2\)

Step 5: Solve for the Variable 'x'

Finally, divide both sides by 2 to solve for \(x\). \(x=1\)