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

Jasonbradley

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Jun 16, 2024

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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.