Answer:
0.469.
Explanation:
To solve the exponential equation 27x = 9^(1/3) * 2^(2/3) * 3^(3/2), we can use the fact that 27 is equal to 3 raised to the power of 3, and 9 is equal to 3 raised to the power of 2. We can also rewrite 2^(2/3) and 3^(3/2) as powers of 2 and 3 respectively.
So, we have:
27x = 3^(3) * 9^(1/3) * 2^(2/3) * 3^(1/2)
27x = 3^(3) * (3^2)^(1/3) * (2^(2))^(1/3) * (3^(2))^(1/4)
27x = 3^(3) * 3^(2/3) * 2^(2/3) * 3^(1/2 * 2)
27x = 3^(3/3 + 2/3 + 1) * 2^(2/3)
27x = 3^(4/3) * 2^(2/3)
Now we can take the logarithm of both sides with base 3:
log₃(27x) = log₃(3^(4/3) * 2^(2/3))
log₃(27x) = 4/3 * log₃(3) + 2/3 * log₃(2)
log₃(27x) = 4/3 + 2/3 * log₃(2)
Simplifying the right-hand side:
log₃(27x) = 2 + 2/3 * log₃(2)
Now we can solve for x by dividing both sides by 27 and using a calculator to evaluate the right-hand side:
log₃(x) = (2 + 2/3 * log₃(2))/27
x = 3^(2 + 2/3 * log₃(2))/27
Using a calculator, we can approximate x to be x ≈ 0.469. Therefore, the solution to the equation 27x = 9^(1/3) * 2^(2/3) * 3^(3/2) is x ≈ 0.469.