(ii) ((-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 problem involves simplifying expressions with exponents and solving for an unknown variable 'x'. Let's break it down step-by-step.

Simplifying the Left-Hand Side

1. Simplify the exponents:

   • ((-3)/(2))^(6) = ( (-3)^6 / (2)^6 ) = (729/64)
   • ((4)/(9))^(3) = ((4)^3 / (9)^3 ) = (64/729)

2. Multiply the simplified terms:

   • (729/64) * (64/729) = 1

Therefore, the left-hand side of the equation simplifies to 1.

Solving for 'x'

1. Rewrite the right-hand side using a common base:

   • ((1)/(2))^(3x) can be written as (2)^(-3x)

2. Equate the simplified expressions:

   • 1 = (2)^(-3x)

3. Solve for 'x':

   • Since any number raised to the power of 0 equals 1, we have:
     • -3x = 0
     • x = 0

Conclusion

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