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 stepbystep.
Simplifying the LeftHand Side

Simplify the exponents:
 ((3)/(2))^(6) = ( (3)^6 / (2)^6 ) = (729/64)
 ((4)/(9))^(3) = ((4)^3 / (9)^3 ) = (64/729)

Multiply the simplified terms:
 (729/64) * (64/729) = 1
Therefore, the lefthand side of the equation simplifies to 1.
Solving for 'x'

Rewrite the righthand side using a common base:
 ((1)/(2))^(3x) can be written as (2)^(3x)

Equate the simplified expressions:
 1 = (2)^(3x)

Solve for 'x':
 Since any number raised to the power of 0 equals 1, we have:
 3x = 0
 x = 0
 Since any number raised to the power of 0 equals 1, we have:
Conclusion
Therefore, the solution to the equation ((3)/(2))^(6)times((4)/(9))^(3)=((1)/(2))^(3x) is x = 0.