Solve each equation. 3^x+2 ⋅ 3^x=81

Verified step by step guidance

1. Recognize that the equation involves exponential expressions with the same base, which is 3: \(3^{x+2} \cdot 3^{x} = 81\).
2. Use the property of exponents that states when multiplying like bases, you add the exponents: \(3^{x+2} \cdot 3^{x} = 3^{(x+2) + x} = 3^{2x+2}\).
3. Rewrite the right side of the equation, 81, as a power of 3. Since \(81 = 3^4\), the equation becomes \(3^{2x+2} = 3^4\).
4. Set the exponents equal to each other because the bases are the same: \(2x + 2 = 4\).
5. Solve the linear equation for \(x\): subtract 2 from both sides to get \(2x = 2\), then divide both sides by 2 to find \(x\).

Key Concepts

Properties of Exponents

This concept involves rules for manipulating expressions with exponents, such as multiplying powers with the same base by adding their exponents. For example, 3^(x+2) * 3^x equals 3^[(x+2) + x] = 3^(2x+2). Understanding these properties simplifies solving exponential equations.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it helps to rewrite constants as powers of the same base as the variable terms. Here, 81 can be expressed as 3^4, allowing the equation to be set with equal bases and exponents, facilitating the solution.

Solving Linear Equations

After equating the exponents, the problem reduces to solving a linear equation in terms of x. This involves isolating x by performing algebraic operations such as addition, subtraction, multiplication, or division to find its value.