[Math] Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$

Question

Which is greater among $300 !$ and $\sqrt {300^{300}}$ ?

The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.

Answer 1

HINT: $(300^{300})^{1/2}=300^{150}$, so you’re comparing

$$\underbrace{300\cdot300\cdot\ldots\cdot300}_{150\text{ factors}}$$

with

$$300\cdot299\cdot\ldots\cdot1=\underbrace{(300\cdot1)\cdot(299\cdot2)\cdot\ldots\cdot(151\cdot150)}_{150\text{ factors}}\;.\tag{1}$$

Show that each of the parenthesized factors in $(1)$ is at least $300$.