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

2 min read Jun 16, 2024
(2/3)^x*(3/2)^2x=81/16 Then X =

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.

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