The expression provided is:

$\sqrt{3^{2a + b}} = \frac{27^a}{3^b}$

We need to express $a$ in terms of $b$.

Step-by-Step Solution:

1. Simplify the Square Root: $\sqrt{3^{2a + b}} = 3^{\frac{2a + b}{2}}$

2. Rewrite $27^a$: Since $27 = 3^3$, we can express $27^a$ as: $27^a = (3^3)^a = 3^{3a}$ So, the equation becomes: $3^{\frac{2a + b}{2}} = \frac{3^{3a}}{3^b}$

3. Simplify the Right-Hand Side: Using the property of exponents $\frac{3^{3a}}{3^b} = 3^{3a - b}$, the equation simplifies to: $3^{\frac{2a + b}{2}} = 3^{3a - b}$

4. Equate Exponents: Since the bases are the same (both are powers of 3), we can equate the exponents: $\frac{2a + b}{2} = 3a - b$

5. Solve for $a$: Multiply both sides of the equation by 2 to eliminate the fraction: $2a + b = 6a - 2b$ Rearrange the equation to isolate $a$: [ 2b + b = 6a - 2a