To evaluate the left-hand side of the equation
\[ x^{\frac{5}{6}} \cdot x^{\frac{3}{2}} = x^a, \]
we can use the property of exponents that states \( x^m \cdot x^n = x^{m+n} \).
- Add the exponents:
\[ \frac{5}{6} + \frac{3}{2}. \]
- To add these fractions, we first need a common denominator. The least common denominator of 6 and 2 is 6. We can convert \(\frac{3}{2}\) to a fraction with a denominator of 6:
\[ \frac{3}{2} = \frac{3 \cdot 3}{2 \cdot 3} = \frac{9}{6}. \]
- Now we can add the two fractions:
\[ \frac{5}{6} + \frac{9}{6} = \frac{5 + 9}{6} = \frac{14}{6}. \]
- Simplify \(\frac{14}{6}\):
\[ \frac{14}{6} = \frac{7}{3}. \]
So, we have
\[ x^{\frac{5}{6}} \cdot x^{\frac{3}{2}} = x^{\frac{7}{3}}. \]
- Therefore, we can equate this to the right-hand side:
\[ x^{a} = x^{\frac{7}{3}}. \]
From this, we can conclude that
\[ a = \frac{7}{3}. \]
Thus, the value of \(a\) in simplest form is
\[ \boxed{\frac{7}{3}}. \]