2000 AMC 10 Problem 2

1. Remember what an exponent means: $2000^{2000}$ is $2000$ copies of the number $2000$ multiplied together.
2. The problem multiplies this by one more $2000$.
3. So now there are $2000 + 1 = 2001$ copies of $2000$ multiplied together, which is $2000^{2001}$.
4. The same thing written with the exponent rule: $2000 = 2000^1$, and when you multiply powers of the same number you add the exponents:
$$2000^{1} \cdot 2000^{2000} = 2000^{2001}.$$

The answer is $\boxed{\textbf{(A)}\ 2000^{2001}}$.

An exponent just counts how many copies are being multiplied. Adding one more copy raises the count by one. You never multiply the bases (that would give $4000$) or multiply the exponents (that only happens for a power of a power, like $(a^2)^3$).

Try the same thing with small numbers where you can check by hand. $2 \cdot 2^3 = 2 \cdot 8 = 16$, and $2^4 = 16$. The wrong patterns give $4^3 = 64$ or $2^6 = 64$, which do not match. The pattern that works is "raise the exponent by one," which is choice (A).

Write the leading $2000$ as $2000^1$. Both factors now share the base $2000$, and powers of the same base multiply by adding exponents:
$$2000^{1} \cdot 2000^{2000} = 2000^{1 + 2000} = 2000^{2001}.$$

The answer is $\boxed{\textbf{(A)}\ 2000^{2001}}$.