Question

Find the greater number in 300! and $\sqrt{{300}^{300}}$

• A. 300!
• B. $\sqrt{{300}^{300}}$
• C. 300! = $\sqrt{{300}^{300}}$
• D. Can't be determined.

Answer 1

The correct option is A

300!

$300!>(100{)}^{300}$
Clearly $(100{)}^{150}>3^{150}$
$\Rightarrow \left(100{)}^{150}\right(100{)}^{150}>3^{150}.(100{)}^{150}\Rightarrow (100{)}^{300}>(300{)}^{150}\Rightarrow (100{)}^{300}>\sqrt{{300}^{300}}$
From (i) and (ii), we get
$300!>(100{)}^{300}>\sqrt{{300}^{300}}Hence300!>\sqrt{{300}^{300}}$
Hence the greater number is 300!.