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]
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]