1. The problem is to find the greatest common divisor (GCD) of $a^n$ and $b^n$. 2. Recall the property of GCD for powers: $$\gcd(a^n,b^n) = (\gcd(a,b))^n$$ 3. This means that the GCD of $a^n$ and $b^n$ is the $n$th power of the GCD of $a$ and $b$. 4. To understand why, consider that any common divisor of $a$ and $b$ raised to the power $n$ will divide both $a^n$ and $b^n$. 5. Conversely, any common divisor of $a^n$ and $b^n$ must be a power of a common divisor of $a$ and $b$. 6. Therefore, the final answer is: $$\boxed{\gcd(a^n,b^n) = (\gcd(a,b))^n}$$