Simplify the expression $$\frac{a^{12} \cdot a^9}{a^{18}}$$ when $$a = 4$$.

Solution:

First, simplify the expression using the properties of exponents. When multiplying powers with the same base, we add the exponents:

\[ \frac{a^{12} \cdot a^9}{a^{18}} = \frac{a^{12+9}}{a^{18}} = \frac{a^{21}}{a^{18}} \]

When dividing powers with the same base, we subtract the exponents:

\[ \frac{a^{21}}{a^{18}} = a^{21-18} = a^3 \]

Now, substitute the given value of $$a = 4$$ into the simplified expression:

\[ a^3 = 4^3 \]

Calculate the value:

\[ 4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64 \]

Answer: 64