So sánh 202 303 và 303 202
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$\begingroup$ Find without use of calculator which of the two numbers is greater $202^{303}$ or $303^{202}$. I think we have to do this with calculus because I got this question from my calculus book. I tried searching on google and SE but didn't get required solution Michael Lugo 22.1k3 gold badges45 silver badges89 bronze badges asked May 15, 2016 at 21:35 $\endgroup$ 6 $\begingroup$ $$\array{808 &\gt& 9 \\ 101 \cdot 2^{3} &\gt& 3^{2} \\ 101^{3} \cdot 2^{3} &\gt& 101^{2} \cdot 3^{2} \\ \left((101\cdot 2){3}\right){101} &\gt& \left((101\cdot 3){2}\right){101} \\ 202^{303} &\gt& 303^{202}}$$ miracle173 11k3 gold badges31 silver badges59 bronze badges answered May 15, 2016 at 22:03 $\endgroup$ 0 $\begingroup$ We have $a^b answered May 15, 2016 at 21:43 Fabio LucchiniFabio Lucchini 16k1 gold badge28 silver badges41 bronze badges $\endgroup$ 0 $\begingroup$ We are comparing $202^{303}$ and $303^{202}$. $202^{303}$ is equal to $202^{202}$ * $202^{101}$. $303^{202}$ is equal to $(202 * 1.5){202}$ which is equal to $202{202}$ * $1.5^{202}$ Now, we can divide out the $202^{202}$ from both sides which yields $202^{101}$ versus $1.5^{202}$. $1.5^{202}$ can be written as $2.25^{101}$ (squaring the inside, thus dividing the exponent by 2). Since $202^{101}$ > $2.25^{101}$, $202^{303}$ > $303^{202}$. No need for calculus! answered May 15, 2016 at 23:55 Mario IshacMario Ishac 2252 silver badges11 bronze badges $\endgroup$ 0 $\begingroup$ $\frac {202^{303}}{303^{202}}=\frac{2^{303}}{3^{202}}\frac{101^{303}}{101^{202}}=\frac{2}{3}{202}*2{101}*101^{101} = (4/3){202} \frac 12{202}*2^{101}*101^{101} = (4/3){202}\frac 12{101}*101^{101} = (4/3){202}*50.5{101} > 1$ So $\frac {202^{303}}{303^{202}} > 1$ (by quite a LOT) So $202^{303} > 303^{202}$ answered May 15, 2016 at 23:46 fleabloodfleablood 123k5 gold badges45 silver badges135 bronze badges $\endgroup$ 0 $\begingroup$ Hint: The function $x \mapsto x^{1/x}$ has a single critical point at $x=e$ and is decreasing for $x \gt e$. answered May 16, 2016 at 0:28 lhflhf 215k19 gold badges235 silver badges550 bronze badges $\endgroup$ $\begingroup$ How to solve in your head: $202 > 2^7, 303 < 2^9$. Therefore $202^{303} > 2^{2121}, 303^{202} < 2^{1818}$. Even easier: $202^{303} > 100^{303} = 1000^{202} > 303^{202}$. Umberto 1,2638 silver badges16 bronze badges answered May 16, 2016 at 11:07 gnasher729gnasher729 9,79418 silver badges35 bronze badges $\endgroup$ $\begingroup$ $$ 202 = 303^{\lg(202) / \lg(303)} \\ 202^{303} = 303^{303 \lg(202)/ \lg(303)} \approx 303^{303\cdot11/12} > 303^{202} $$ answered May 16, 2016 at 12:46 Olba12Olba12 2,5412 gold badges16 silver badges29 bronze badges $\endgroup$ $\begingroup$ $$202^{303} = \left(\frac{2}{3}\right){303} \times 303{303}= \left(\frac{2}{3}\right){303} \times 303{101} \times 303^{202}$$ $$= \left(\left(\frac{2}{3}\right)3\right){101}\times 303^{101}\times 303^{202}$$ $$=\left(\frac{8 \times 303}{27}\right){101} \times 303{202}$$ $$=\left(\frac{808}{9}\right){101} \times 303{202} > 303^{202}$$ answered May 9, 2018 at 20:39 John_dydxJohn_dydx 4,1801 gold badge18 silver badges28 bronze badges $\endgroup$ $\begingroup$ $(202){303}=(2\times 101){3\times {101}}=2^{3\times 101} \times 101^{303}=8^{101}\times 101^{303}$ $(303){202} = (3 \times 101){2 \times 101} = 3^{2\times 101} \times 101^{202}=9^{101}\times 101^{202}$ So $$\frac{(202){303}}{(303){202}}= \frac{8^{101}\times 101^{303}}{9^{101}\times 101^{202}}= \frac{8^{101}}{9^{101}}\times 101^{101}=\left(\frac{808}{9}\right)^{101}$$ |
