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Solving for x: (2/3)^x * (3/2)^2x = 81/16
This problem involves simplifying exponents and solving an exponential equation. Let's break down the steps:
1. Simplify the exponents:
- (3/2)^2x = (3^2x / 2^2x) = (9^x / 4^x)
2. Rewrite the equation:
- Now the equation becomes: (2/3)^x * (9^x / 4^x) = 81/16
3. Combine terms with the same base:
- (2^x * 9^x) / (3^x * 4^x) = 81/16
- (18^x) / (12^x) = 81/16
4. Simplify further:
- (3^x * 6^x) / (3^x * 4^x) = 81/16
- (6^x) / (4^x) = 81/16
- (3^x * 2^x) / (2^x * 2^x) = 81/16
- (3^x / 2^x) = 81/16
5. Express both sides with the same base:
- (3/2)^x = (3/2)^4
6. Solve for x:
- Since the bases are the same, we can equate the exponents:
- x = 4
Therefore, x = 4.