Which is larger $\sqrt[99]{99!}$ or $\sqrt[100]{100!}$
Solution 1:
Notice that $\sqrt[99]{99!}$ is the geometric mean of all the integers from 1 to 99, while $\sqrt[100]{100!}$ is the geometric mean of those same integers and 100. What happens to a mean when you include one more element that is larger than all the previous ones?
Solution 2:
If you bring both to the $99*100$ power, you get the two numbers$$(99!)^{100},(100!)^{99}=(99!)^{99}100^{99}$$ So dividing out the common factor we want to compare $$99!,100^{99}$$ It should be clear which is larger.