Đáp án + Giải thích các bước giải:

 1) Ta có:
$10^{30} = (10^3)^{10} = 1000^{10}$

$2^{100} = (2^10)^{10} = 1024^{10}$

Mà $1000^{10} < 1024^{10}$

$\Rightarrow 10^{30} < 2^{100}$
2) Ta có:

$333^{444} = (333^4)^{111} = (111^4 . 3^4)^{111} = (111^4 . 81)^{111}$
$444^{333} = (444^3)^{111} = (111^4 . 4^3)^{111} = (111^4 . 64)^{111}$

Mà $(111^4 . 81)^{111} > (111^4 . 64)^{111}$

$\Rightarrow 333^{444} > 444^{333}$
3) Ta có:
$2^{161} > 2^{160} = (2^4)^{40} = 16^{40}$
Mà $16^{40} > 13^{40}$

$\Rightarrow 2^{161} > 13^{40}$
4) Ta có:
$5^{300} = (5^2)^{150} = 25^{150}$
$3^{453} > 3^{450} = (3^3)^{150} = 27^{150}$
Mà $25^{150} < 27^{150}$

$\Rightarrow 5^{300} < 3^{453}$

5) Ta có:
$5^{217} > 5^{216} = (5^3)^{72} = 125^{72}$
Mà $125^{72} > 119^{72}$
$\Rightarrow 5^{217} > 119^{72}$
6) Ta có:
$2^{100} = (2^{10})^{10} = 1024^{10}$

Mà $1024^{10} > 1024^9$

$\Rightarrow 2^{100} > 1024^9$