Find the greater number in 300 ! and
`sqrt(300^(sqrt(300)))`

Explanation

Since , `(100)^(150) gt 3^(150)`
` rArr (100)^(150) . (100)^(150) gt 3^(150) .(100)^(150)`
` rArr (100)^(100) gt (300)^(150)`
or ` (100)^(300)gt (300)^(150)` …(i)
Using result (ii) ` ((n)/(3))^(n) lt n !`
Putting n = 300, we get ` (100)^(300) lt 300! `
From Eqs . (i) and (ii) , we get
`sqrt(300^(300)) lt (100)^(300) lt 300!`
` rArr sqrt(300^(300)) lt 300! `
Hence , the greater number is 300 ! .