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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
  

Given that \(\sqrt{3^{2a+b}} = \frac{27^a}{3^b}\), express \(a\) in terms of \(b\).

Explore how to solve the equation \(\sqrt{3^{2a+b}} = \frac{27^a}{3^b}\) by expressing \(a\) in terms of \(b\). This problem involves algebraic manipulation and understanding the properties of exponents and square roots, suitable for students in Grades 9-12.

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