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Question Number 191443 by MATHEMATICSAM last updated on 24/Apr/23

$$(3^{100}+2^{100}),4^{100}\mathrm{which}\mathrm{is}\mathrm{greater}?$$$$\mathrm{Give}\mathrm{proof}.$$

Answered by Frix last updated on 24/Apr/23

$$4^{100}=2^{200}>2^{100}+3^{100}$$$$2^{200}-2^{100}>3^{100}$$$$2^{100}(2^{100}-1)>3^{100}$$$$2^{100}-1>{\left(\frac{3}{2}\right)}^{100}\mathrm{which}\mathrm{is}\mathrm{obvious}$$

Answered by mehdee42 last updated on 24/Apr/23

$$4^{100}={16}^{50}>{13}^{50}={(4+9)}^{50}>4^{50}+9^{50}$$$$\Rightarrow 4^{100}>2^{100}+3^{100}$$

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