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(125/27)x(125/27)^x=(5/3)^18

2 min read Jun 16, 2024
(125/27)x(125/27)^x=(5/3)^18

Solving the Equation (125/27)x(125/27)^x = (5/3)^18

This article will guide you through solving the equation (125/27)x(125/27)^x = (5/3)^18. We'll use a combination of algebraic manipulation and understanding of exponents to find the solution for x.

Simplifying the Equation

  1. Rewrite the equation with the same base: Notice that both (125/27) and (5/3) can be expressed as powers of (5/3).

    • (125/27) = (5/3)^3
    • (5/3) = (5/3)^1

    Now, substitute these values back into the original equation: (5/3)^3 * (5/3)^(3x) = (5/3)^18

  2. Apply exponent rules: When multiplying exponents with the same base, we add the powers. Therefore, the equation simplifies to: (5/3)^(3 + 3x) = (5/3)^18

  3. Equate the exponents: Since the bases are equal, we can equate the exponents: 3 + 3x = 18

Solving for x

  1. Isolate x: Subtract 3 from both sides of the equation: 3x = 15

  2. Solve for x: Divide both sides by 3: x = 5

Solution

Therefore, the solution to the equation (125/27)x(125/27)^x = (5/3)^18 is x = 5.

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