[Not-so-daily puzzle] Which one is bigger? \ stacker news ~science

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Evaluate, without using a calculator, which one is bigger:

 $10^{36} \text{ or } 9^{37}\ ?$

Sats will be given for each different approach.

Previous iteration: #815243 (fun variant in #815534).

 $10^{36} > 9^{37}$

 $10^{36} > 9 * 9^{36}$

 $\frac{10^{36}}{9^{36}} > 9$

 $(\frac{10}{9})^{36} > 9$

 $(1 + \frac{1}{9})^{36} > 9$

 $((1 + \frac{1}{9})^9)^4 > 9$

(1+x)^r \geq 1+rx

to the inner term

 $(1+\frac{1}{9})^9 \geq 1+9*\frac{1}{9}$

 $(1+\frac{1}{9})^9 \geq 2$

Inserting into the outer term

 $2^4 > 9$

I think it's

10^{36} > 9^{37}

Obviously,

10^{36} > 9^{36}

If you multiply the RHS by

(10/9)^{36}

it will equal the LHS and that multiplier is greater than 9 (I'm pretty sure).

.../watch?v=Opp1EDSC-Ho

Did not use a calculator, but used ChatGPT:

10^36 is bigger.

My approach is very childish like i imagined having stacks of 10, 36 layer higher. And stack of 9s , 37 layer higher.

I hope I'm not wrong lol

Take heed, a characteristic single sat zap was just received. @Scroogey's neurons are on the case...