Solve the given equation. \( \left(4^{\frac{x}{2}}\right)\left(4^{\frac{x}{3}}\right)=4^{9} \)

To solve the equation \( \left(4^{\frac{x}{2}}\right)\left(4^{\frac{x}{3}}\right)=4^{9} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). So, we add the exponents on the left-hand side: \[ 4^{\frac{x}{2}+\frac{x}{3}} = 4^{9} \] Next, we need to combine the exponents \( \frac{x}{2} \) and \( \frac{x}{3} \): To do this, we need a common denominator. The least common multiple of 2 and 3 is 6. Thus, we can write: \[ \frac{x}{2} = \frac{3x}{6} \quad \text{and} \quad \frac{x}{3} = \frac{2x}{6} \] Adding these gives: \[ \frac{x}{2} + \frac{x}{3} = \frac{3x}{6} + \frac{2x}{6} = \frac{5x}{6} \] Now we rewrite the equation as: \[ 4^{\frac{5x}{6}} = 4^{9} \] Since the bases are the same, we can set the exponents equal to each other: \[ \frac{5x}{6} = 9 \] To solve for \( x \), multiply both sides by 6: \[ 5x = 54 \] Now divide both sides by 5: \[ x = \frac{54}{5} \] Therefore, the solution to the equation is: \[ x = 10.8 \]