What is the quotient of $50^{50}$ by $25^{25}$?
(a) $25^{25}$
(b) $10^{25}$
(c) $100^{25}$
(d) $2^{25}$
(e) $2 \times 25^{25}$
The correct option is (c).

Solution 1: $\frac{50^{50}}{25^{25}}=\frac{\left(2 \times 5^{2}\right)^{50}}{\left(5^{2}\right)^{25}}=\frac{2^{50} \times 5^{100}}{5^{50}}=2^{50} \times 5^{50}=\left(2^{2} \times 5^{2}\right)^{25}=100^{25}$.

Solution 2: $\frac{50^{50}}{25^{25}}=\frac{(2 \times 25)^{50}}{25^{25}}=\frac{2^{50} \times 25^{50}}{25^{25}}=2^{25} \times 2^{25} \times 25^{25}=100^{25}$.