Give the greatest common divisor of $6^3$ and $3^6$.
I need to find the largest number that divides both $6^3$ and $3^6$ evenly. One way to do this is to factor both numbers into prime factors and look for the common ones. For $6^3$, I can use the fact that $6 = 2 \times 3$ and apply the exponent rule $(ab)^n = a^n b^n$. So, $6^3 = (2 \times 3)^3 = 2^3 \times 3^3 = 8 \times 27$. For $3^6$, I can use the fact that $3$ is already prime and apply the exponent rule $a^n = a \times a \times \cdots \times a$. So, $3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$. Now, I can see that both $6^3$ and $3^6$ have $3^3 = 27$ as a common factor, but nothing else. Therefore, the greatest common divisor of $6^3$ and $3^6$ is $3^3 = 27$. 