Câu hỏi của Red Brown

Question

So sánh $2^{30}+3^{30}+4^{30}$ và $3.24^{10}$

Le Thi Khanh Huyen · Accepted Answer

Ta có:

$3.24^{10}=3.\left(2^3.3\right)^{10}=3.2^{30}.3^{10}=3^{11}.2^{30}$

$4^{30}=\left(2^2\right)^{30}=2^{60}=2^{30}.2^{30}=4^{15}.2^{30}$

Dễ thấy $3^{11}.2^{30}< 4^{15}.2^{30}$

$\Rightarrow3.24^{10}< 4^{30}< 2^{30}+3^{30}+4^{30}$

Vậy $3.24^{10}< 2^{30}+3^{30}+4^{30}$

Câu trả lời:

Ta có:

$3.24^{10}=3.\left(2^3.3\right)^{10}=3.2^{30}.3^{10}=3^{11}.2^{30}$

$4^{30}=\left(2^2\right)^{30}=2^{60}=2^{30}.2^{30}=4^{15}.2^{30}$

Dễ thấy $3^{11}.2^{30}< 4^{15}.2^{30}$

$\Rightarrow3.24^{10}< 4^{30}< 2^{30}+3^{30}+4^{30}$

Vậy $3.24^{10}< 2^{30}+3^{30}+4^{30}$