Which of the following numbers is the smallest: \(2^{250}, 3^{150}, 5^{100}, 4^{200}\)

Compare by writing all numbers in terms of prime bases.

$4^{200} = (2^2)^{200} = 2^{400}$

Now we compare:
$2^{250},\ 2^{400},\ 3^{150},\ 5^{100}$

Clearly, $2^{400}$ is larger than $2^{250}$, so ignore it.

Now compare $3^{150}$ and $5^{100}$ with $2^{250}$.

Take logarithm idea (or compare growth):

$3^{150} = (3^3)^{50} = 27^{50}$
$5^{100} = (5^2)^{50} = 25^{50}$

So, $3^{150} > 5^{100}$

Now compare $2^{250}$ and $5^{100}$:

$2^{250} = (2^5)^{50} = 32^{50}$
$5^{100} = 25^{50}$

So, $2^{250} > 5^{100}$

Hence, the smallest number is $5^{100}$.