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Question Number 137065 by mr W last updated on 29/Mar/21

Commented by mr W last updated on 29/Mar/21

which number is larger?

$${which}\:{number}\:{is}\:{larger}? \\ $$

Answered by mr W last updated on 30/Mar/21

3^2 =9>8.8=1.1×8=(1+0.1)×2^3   (3^2 )^(50) >(1+0.1)^(50) ×(2^3 )^(50)   3^(100) >(1+0.1)^(50) ×2^(150) >(1+50×0.1)×2^(150) =6×2^(150)   (1/2)×3^(100) >3×2^(150) >2^(150)   2^2 =4>3  (2^2 )^((1/2)×3^(100) ) >3^2^(150)    ⇒2^3^(100)  >3^2^(150)

$$\mathrm{3}^{\mathrm{2}} =\mathrm{9}>\mathrm{8}.\mathrm{8}=\mathrm{1}.\mathrm{1}×\mathrm{8}=\left(\mathrm{1}+\mathrm{0}.\mathrm{1}\right)×\mathrm{2}^{\mathrm{3}} \\ $$$$\left(\mathrm{3}^{\mathrm{2}} \right)^{\mathrm{50}} >\left(\mathrm{1}+\mathrm{0}.\mathrm{1}\right)^{\mathrm{50}} ×\left(\mathrm{2}^{\mathrm{3}} \right)^{\mathrm{50}} \\ $$$$\mathrm{3}^{\mathrm{100}} >\left(\mathrm{1}+\mathrm{0}.\mathrm{1}\right)^{\mathrm{50}} ×\mathrm{2}^{\mathrm{150}} >\left(\mathrm{1}+\mathrm{50}×\mathrm{0}.\mathrm{1}\right)×\mathrm{2}^{\mathrm{150}} =\mathrm{6}×\mathrm{2}^{\mathrm{150}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{3}^{\mathrm{100}} >\mathrm{3}×\mathrm{2}^{\mathrm{150}} >\mathrm{2}^{\mathrm{150}} \\ $$$$\mathrm{2}^{\mathrm{2}} =\mathrm{4}>\mathrm{3} \\ $$$$\left(\mathrm{2}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{3}^{\mathrm{100}} } >\mathrm{3}^{\mathrm{2}^{\mathrm{150}} } \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{3}^{\mathrm{100}} } >\mathrm{3}^{\mathrm{2}^{\mathrm{150}} } \\ $$

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