Which is greater? Ⅰ 95％ fail to solve!

Which is larger, 10^36 or 9^37? How do you solve this equation? Leran the wonderful method.

[doronezri]

The lower bound of 2 is great! But hard to get to (independently)... You can get the same bound by taking the first to terms of the Binomial expansion of (1+1/9)^9 😊

[pietergeerkens]

Nice problem.

Squaring 0.729 = 0.9^3 to get 0.531441 = 0.9^6 shows that
0.9^7 < 1/2
and
0.9^35 = (0.9^7)^5 < (1/2)^5 = 1/32
since


Thus
9^37 / 10^36 = 9 * 9/10 * (0.9)^35 < 81/10 * 1/32 = 81/320 << 1.

Hence 10^36 is bigger.

[valdimer]

Ive never seen an integer smaller with a smaller exponent.

[hobbified]

9^37 = 9 * 9^36 = 9 * (.9)^36 * 10^36. (.9)^36 is definitely a LOT less than 1/9.

[rabotaakk-nwnm]

2:25-4:50 n>1 => 2 < (1+1/n)ⁿ < e

[pietersfilms]

10^37 or 10*9^37

37 or 1+37*log(9)
36 or 37*log(9)

Hmmm try again

10 or 9*10^1/37
Uhh

[yurenchu]

My gut feeling says 10³⁶ > 9³⁷, because for large n, (1 + 1/n)ⁿ ≈ e which means

(10/9)³⁶ = [ (1 + 1/9)⁹ ]⁴ ≈ [e]⁴ > 2⁴ = 16 > 9

and hence 10³⁶ = (10/9)³⁶ * 9³⁶ ≈ e⁴ * 9³⁶ > 9 * 9³⁶ = 9³⁷ .

But to prove it rigorously:

(10/9)⁹ =
= (1 + 1/9)⁹
= Σ B(9, k) * (1/9)ᵏ, from k=0 to k=9
... where _binomial coefficient_ B(9, k) = (9 choose k) = 9!/[k! * (9-k)!] ...
= B(9, 0) + B(9, 1)*(1/9)¹ + B(9, 2)*(1/9)² + B(9, 3)*(1/9)³ + B(9, 4)*(1/9)⁴ + ...
= [1] + [9]*(1/9) + [36]*(1/81) + [84]*(1/729) + [126]*(1/6561) + ...
... note the higher terms are all positive, and hence ...
> 1 + 9/9 + 36/81 + 84/729 + 126/6561
= 1 + 1 + 324/729 + 84/729 + 14/729
= 2 + 422/729
> 2.5

and therefore,

10³⁶ =
= (10/9)³⁶ * 9³⁶
= [ (10/9)⁹ ]⁴ * 9³⁶
> [ 2.5 ]⁴ * 9³⁶
> 2⁴ * 9³⁶
= 16 * 9³⁶
> 9 * 9³⁶
= 9³⁷

[somebodysomewhere]

You literally just take the binominal expainsion of (9+1)^36, then its clear that there more than 9 times 9^36 in there.

[alex.harrison]

[Ln(10)/ln(9) x 36] > 37 (actually ~=37.73)

[Speedcraft-oi]

10^36 9^37
10^36 9^36*9
10 and 9*9^1/36
*remember for every number under 10 after or the 22th root is less then 10/9.*
10 and 9*(less then 10/9)
10 is bigger than 9*(less then 10/9)
∵10 is bigger
∴so 10^36 > 9^37