Answer:
To find an expression for x in terms of a and b, we need to simplify the given equation and isolate x.
Let's start by rewriting the equation using the properties of exponents:
9^x = (27^a)^(1/2) * 3^b
Since (27^a)^(1/2) is equal to the square root of 27^a, we can simplify further:
9^x = √(27^a) * 3^b
Now, let's simplify the expression inside the square root:
√(27^a) = √((3^3)^a) = √(3^(3a)) = (3^(3a))^(1/2) = 3^(3a/2)
Plugging this back into the equation, we have:
9^x = 3^(3a/2) * 3^b
To combine the terms on the right side of the equation, we add the exponents of 3:
9^x = 3^(3a/2 + b)
Now, we can rewrite 9^x as (3^2)^x = 3^(2x) to match the base:
3^(2x) = 3^(3a/2 + b)
Since the bases are the same, the exponents must be equal:
2x = 3a/2 + b
Finally, we can solve for x by dividing both sides of the equation by 2:
x = (3a/2 + b) / 2
Therefore, the expression for x in terms of a and b is:
x = (3a/2 + b) / 2
I hope this explanation helps!
Explanation: