To simplify this expression, we can first try to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator:

[tex]\frac{a\sqrt{a}+27 }{a-3\sqrt{a}+9} \cdot \frac{a+3\sqrt{a}+9}{a+3\sqrt{a}+9}[/tex]

Multiplying the numerators and denominators, we get:

[tex]\frac{(a\sqrt{a}+27)(a+3\sqrt{a}+9)}{(a-3\sqrt{a}+9)(a+3\sqrt{a}+9)}[/tex]

Now, let's expand the numerator:

[tex]a^2 + 3a\sqrt{a} + 9a + 3a\sqrt{a} + 9\sqrt{a} + 27 = a^2 + 6a\sqrt{a} + 9a + 9\sqrt{a} + 27[/tex]

And expand the denominator:

[tex]a^2 + 3a\sqrt{a} + 9a - 3a\sqrt{a} - 9\sqrt{a} + 9a - 3a\sqrt{a} - 9\sqrt{a} + 27 = a^2 + 6a\sqrt{a} + 9a[/tex]

Therefore, the simplified form of the expression is:

[tex]\frac{a^2 + 6a\sqrt{a} + 9a + 9\sqrt{a} + 27}{a^2 + 6a\sqrt{a} + 9a}[/tex]