(2/3)^x*(3/2)^2x=81/16

Jasonbradley

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Jun 16, 2024

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Solving the Exponential Equation: (2/3)^x * (3/2)^2x = 81/16

This article will guide you through the process of solving the exponential equation (2/3)^x * (3/2)^2x = 81/16. We'll utilize the properties of exponents and simplification techniques to arrive at the solution.

Understanding the Problem

We have an equation with a variable 'x' in the exponent. Our goal is to isolate 'x' and find its value.

Solving the Equation

1. Simplifying the Left-Hand Side:

   • Using the property (a/b)^m = a^m / b^m, we can rewrite the equation as:

     (2^x / 3^x) * (3^(2x) / 2^(2x)) = 81/16

   • Further simplification by combining terms with the same base:

     2^(x-2x) * 3^(2x-x) = 81/16 2^(-x) * 3^x = 81/16

2. Expressing both sides with the same base:

   • We notice that 81 and 16 can be expressed as powers of 3 and 2 respectively:

     2^(-x) * 3^x = 3^4 / 2^4

3. Equating the exponents:

   • For the equation to hold true, the exponents of both sides must be equal:

     -x = -4 and x = 4

4. Solution:

   • Solving for 'x' we get:

     x = 4

Conclusion

We have successfully solved the equation (2/3)^x * (3/2)^2x = 81/16 and found the value of x to be 4. This solution demonstrates the application of exponent properties and algebraic manipulation to solve exponential equations.