(a) Expand the following. $$ a^{3}-729 $$

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To expand $$ a^{3} - 729 $$, we can recognize that $$ 729 $$ is a perfect cube, specifically $$ 729 = 9^3 $$. Therefore, we can rewrite the expression as:

$$
a^{3} - 9^{3}
$$

This expression can be factored using the difference of cubes formula, which states that:

$$
x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2})
$$

In our case, let $$ x = a $$ and $$ y = 9 $$. Applying the formula, we have:

$$
a^{3} - 9^{3} = (a - 9)(a^{2} + a \cdot 9 + 9^{2})
$$

Now, we can simplify the second factor:

$$
9^{2} = 81
$$

Thus, the expression becomes:

$$
a^{3} - 729 = (a - 9)(a^{2} + 9a + 81)
$$

So, the expanded form of $$ a^{3} - 729 $$ is:

$$
(a - 9)(a^{2} + 9a + 81)
$$
The expanded form of $$ a^{3} - 729 $$ is $$ (a - 9)(a^{2} + 9a + 81) $$.

(a) Expand the following. \( a^{3}-729 \)

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