\( \frac{2x^{\frac{1}{2}}.x^{\frac{2}{3}}.6x^{\frac{-7}{3}}}{9x^{\frac{-5}{3}}.x^{\frac{3}{2}}} \)

To simplify the expression \( \frac{2x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} \cdot 6x^{\frac{-7}{3}}}{9x^{\frac{-5}{3}} \cdot x^{\frac{3}{2}}} \), first, we can combine the constants and the exponents for \( x \). Step 1: Combine constants. \[ \frac{2 \cdot 6}{9} = \frac{12}{9} = \frac{4}{3} \] Step 2: Combine exponents of \( x \) in the numerator and the denominator. \[ x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} \cdot x^{\frac{-7}{3}} = x^{\left(\frac{1}{2} + \frac{2}{3} - \frac{7}{3}\right)} \] To simplify the exponent, find a common denominator, which is 6: \[ \frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}, \quad -\frac{7}{3} = -\frac{14}{6} \] Now, adding up the exponents: \[ \frac{3}{6} + \frac{4}{6} - \frac{14}{6} = \frac{3 + 4 - 14}{6} = \frac{-7}{6} \] So, the numerator becomes: \[ x^{\frac{-7}{6}} \] Step 3: Simplifying the denominator: \[ x^{\frac{-5}{3}} \cdot x^{\frac{3}{2}} = x^{\left(-\frac{5}{3} + \frac{3}{2}\right)} \] Finding common denominators again: \[ -\frac{5}{3} = -\frac{10}{6}, \quad \frac{3}{2} = \frac{9}{6} \] Thus, \[ -\frac{10}{6} + \frac{9}{6} = -\frac{1}{6} \] So the denominator becomes: \[ x^{-\frac{1}{6}} \] Step 4: Finally, combining the combined parts: \[ \frac{4}{3} \cdot \frac{x^{\frac{-7}{6}}}{x^{-\frac{1}{6}}} = \frac{4}{3} \cdot x^{\left(-\frac{7}{6} + \frac{1}{6}\right)} = \frac{4}{3} \cdot x^{-\frac{6}{6}} = \frac{4}{3} \cdot x^{-1} \] This can also be written as: \[ \frac{4}{3x} \] The simplified expression is: \[ \frac{4}{3x} \]