2. Solve the equation \( \left(3^{\frac{x}{2}}\right)\left(3^{\frac{x}{4}}\right)=3^{6} \). \( x=\square \)

To solve the equation \( \left(3^{\frac{x}{2}}\right)\left(3^{\frac{x}{4}}\right)=3^{6} \), first, use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Combining the exponents gives: \[ 3^{\frac{x}{2} + \frac{x}{4}} = 3^{6} \] Next, find a common denominator for the exponents on the left side. The common denominator of 2 and 4 is 4: \[ \frac{x}{2} = \frac{2x}{4} \] \[ \frac{x}{4} = \frac{x}{4} \] Adding these, we have: \[ \frac{2x}{4} + \frac{x}{4} = \frac{3x}{4} \] So the equation simplifies to: \[ 3^{\frac{3x}{4}} = 3^{6} \] Since the bases are the same, you can set the exponents equal to each other: \[ \frac{3x}{4} = 6 \] To isolate \( x \), multiply both sides by \( \frac{4}{3} \): \[ x = 6 \cdot \frac{4}{3} = 8 \] Thus, the solution is \( x = 8 \).