We are given the equation:

$a + \frac{9}{a} = 3$

Step 1: Multiply both sides by $a$ to eliminate the fraction:

$a^2 + 9 = 3a$

Rearrange the equation:

$a^2 - 3a + 9 = 0$

Step 2: Express $a^3 + 27$

We recognize that:

$a^3 + 27 = (a+3)(a^2 - 3a + 9)$

From Step 1, we know:

$a^2 - 3a + 9 = 0$

So,

$(a+3) \times 0 = 0$

Thus,

$a^3 + 27 = \mathbf{0}$

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Final Answer:

$\mathbf{0}$

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Do you want more details? Here are some related questions:

1. How do we solve similar equations with fractions?
2. What is the factorization of $a^3 + 27$?
3. How does this relate to the sum and product of roots?
4. Can we solve $a^3 + 27$ using another method?
5. What happens if we substitute different values for $a$?

Tip: Always check if an equation can be factored or rewritten before solving directly.