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(1/3)^x+3+(1/3)^x+2=4/27

2 min read Jun 16, 2024
(1/3)^x+3+(1/3)^x+2=4/27

Solving the Equation (1/3)^x+3 + (1/3)^x+2 = 4/27

This article will guide you through solving the exponential equation (1/3)^x+3 + (1/3)^x+2 = 4/27. We'll break down the steps and use key properties of exponents to find the solution.

Understanding the Equation

The equation involves terms with the base (1/3) raised to different powers. To solve it effectively, we need to manipulate the equation using the properties of exponents.

Step-by-Step Solution

  1. Simplify the equation:

    • Notice that both terms on the left-hand side have (1/3)^x as a common factor.
    • Rewrite the equation as: (1/3)^x * (1/3)^3 + (1/3)^x * (1/3)^2 = 4/27
    • Simplify further: (1/3)^x * (1/27) + (1/3)^x * (1/9) = 4/27

  2. Factor out (1/3)^x:

    • (1/3)^x * [(1/27) + (1/9)] = 4/27

  3. Combine the constants:

    • (1/3)^x * (4/27) = 4/27

  4. Isolate (1/3)^x:

    • (1/3)^x = 1

  5. Express 1 as a power of (1/3):

    • (1/3)^x = (1/3)^0

  6. Solve for x:

    • Since the bases are the same, we can equate the exponents: x = 0

Conclusion

The solution to the equation (1/3)^x+3 + (1/3)^x+2 = 4/27 is x = 0. By simplifying the equation and utilizing the properties of exponents, we were able to isolate the variable and solve for its value.

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