Evaluate the expression \(\frac{a^3 + 27}{a^2 - 3a + 9}\) for \(a = 15\).

Solution:

The expression is \( \frac{a^3 + 27}{a^2 - 3a + 9} \). We can factor the numerator as a sum of cubes: \(a^3 + 3^3 = (a+3)(a^2 - 3a + 9)\).

So, the expression becomes \( \frac{(a+3)(a^2 - 3a + 9)}{a^2 - 3a + 9} \).

For \(a
eq 3\), we can cancel out the term \(a^2 - 3a + 9\), which simplifies the expression to \(a+3\).

Now, substitute \(a = 15\):

\[ 15 + 3 = 18 \]

Answer: 18