2.1 Simplify fully, WITHOUT using a calculator: \( \frac{3^{2 x+1} \cdot 15^{2 x-3}}{27^{x-1} \cdot 3^{x} \cdot 5^{2 x-1}} \)

To simplify the expression \( \frac{3^{2x+1} \cdot 15^{2x-3}}{27^{x-1} \cdot 3^{x} \cdot 5^{2x-1}} \), we start by rewriting the bases involved: 1. Notice that \( 15 = 3 \cdot 5 \), hence \( 15^{2x-3} = (3 \cdot 5)^{2x-3} = 3^{2x-3} \cdot 5^{2x-3} \). 2. The term \( 27 \) can be expressed as \( 3^3 \), so \( 27^{x-1} = (3^3)^{x-1} = 3^{3(x-1)} = 3^{3x-3} \). Substituting these into the original expression, we have: \[ \frac{3^{2x+1} \cdot 3^{2x-3} \cdot 5^{2x-3}}{3^{3x-3} \cdot 3^x \cdot 5^{2x-1}} \] Next, we can combine similar bases: \[ = \frac{3^{(2x+1) + (2x-3)} \cdot 5^{2x-3}}{3^{(3x-3) + x} \cdot 5^{2x-1}} \] Now simplifying the exponents for \(3\): \[ = \frac{3^{4x - 2} \cdot 5^{2x-3}}{3^{4x - 3} \cdot 5^{2x-1}} \] This can be simplified by subtracting the exponents of \(3\) and \(5\): For \(3\): \[ 3^{(4x - 2) - (4x - 3)} = 3^{1} = 3 \] For \(5\): \[ 5^{(2x - 3) - (2x - 1)} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \] Putting it all together, we get: \[ \frac{3}{25} \] Thus, the fully simplified expression is: \[ \boxed{\frac{3}{25}} \]