((-3)/(2))^(6)times((4)/(9))^(3)=((1)/(2))^(3x)

less than a minute read Jun 16, 2024
((-3)/(2))^(6)times((4)/(9))^(3)=((1)/(2))^(3x)

Solving the Equation: ((-3)/(2))^(6) times ((4)/(9))^(3) = ((1)/(2))^(3x)

This equation involves exponents and fractions, requiring us to apply various exponent rules to solve for the unknown variable 'x'. Let's break down the solution step by step:

Simplifying the Left Hand Side

  1. Simplify the exponents:

    • ((-3)/(2))^(6) = (-3/2)^6 = 729/64
    • ((4)/(9))^(3) = (4/9)^3 = 64/729
  2. Multiply the simplified terms:

    • (729/64) * (64/729) = 1

Now the equation becomes: 1 = ((1)/(2))^(3x)

Solving for x

  1. Recognize that 1 is any number raised to the power of 0:

    • 1 = (1/2)^0
  2. Equate the exponents:

    • (1/2)^0 = (1/2)^(3x)
    • 0 = 3x
  3. Solve for x:

    • x = 0/3
    • x = 0

Therefore, the solution to the equation ((-3)/(2))^(6) times ((4)/(9))^(3) = ((1)/(2))^(3x) is x = 0.

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