Which number is larger, $2^{300}$ or $3^{200}$? Well, calculating such numbers explicitly is quite difficult, even with the help of a calculator or a computer. However, we can figure it out without explicitly calculating them! Notice:

$$
2^{300}=2^{3 \cdot 100}=\left(2^{3}\right)^{100}=8^{100}
$$

and also notice that

$$
3^{200}=3^{2 \cdot 100}=\left(3^{2}\right)^{100}=9^{100}
$$

Since $8^{100}<9^{100}$, we conclude that $2^{300}<3^{200}$.

a) Which number is larger, $2^{40}$ or $3^{28}$?

b) Which number is larger, $31^{11}$ or $17^{14}$?
Solution

a) To see which number is larger, we do:

$$
\begin{aligned}
& 2^{40}=2^{3 \cdot 13+1}=2 \cdot 2^{3 \cdot 13}=2 \cdot\left(2^{3}\right)^{13}=2 \cdot 8^{13} \\
& 3^{28}=3^{2 \cdot 14}=\left(3^{2}\right)^{14}=9^{14}=9 \cdot 9^{13}
\end{aligned}
$$

Since $2 \cdot 8^{13}<9 \cdot 9^{13}$, we conclude that $2^{40}<3^{28}$.

b) To see which number is larger, we do:

$$
31^{11}<32^{11}=\left(2^{5}\right)^{11}=2^{55}<2^{56}=\left(2^{4}\right)^{14}<17^{14}
$$