7. Which of these numbers is the largest?
A $2^{5000}$
B $3^{4000}$
C $4^{3000}$
D $5^{2000}$
E $6^{1000}$
SolUTION
B

We have
$$
\begin{array}{l}
2^{5000}=\left(2^{5}\right)^{1000}=32^{1000} \\
3^{4000}=\left(3^{4}\right)^{1000}=81^{1000}, \\
4^{3000}=\left(4^{3}\right)^{1000}=64^{1000}
\end{array}
$$
and
$$
5^{2000}=\left(5^{2}\right)^{1000}=25^{1000} .
$$

Since $6<25<32<64<81$, it follows that $6^{1000}<25^{1000}<32^{1000}<64^{1000}<81^{1000}$. Therefore $6^{1000}<5^{2000}<2^{5000}<4^{3000}<3^{4000}$.
Hence, of the given numbers, it is $3^{4000}$ that is the largest.