# I lost my calculator batteries in a flood .

1. Aug 6, 2004

### BobG

I lost my calculator batteries in a flood.....

What's this equal?

$$\left(\frac{256^{16}-1}{256^{16}}\right)^{256^{16}}$$

a) 2.178
b) 1.000
c) .3679
d) 0.000

Last edited: Aug 6, 2004
2. Aug 6, 2004

### chroot

Staff Emeritus
Well, it's obviously not one, because the number inside the parentheses is not one.

It's also obviously not zero for the same reason.

It's not 2.178, because the number inside the parentheses is less than one, and multiplying a fraction smaller than one by another fraction smaller than one must result in a fraction smaller than one.

By the process of elimination, it must be 0.3679.

- Warren

3. Aug 6, 2004

### Manu2380

i get 1/e..........i could be missing something in my work.....or maybe even forgetting my calculus lol...

y = [(x-1)/x]^x
lny = x[ln(x-1) - lnx]
lny = ln(x-1 / x) / 1/x
using lhopitals....

lny = (x/(x-1))(1/x^2) / -1/x^2
lny = -x/ x-1 as x approaches infinity......lny = -1 so y = 1/e

But i may be wrong since i am assuming x goes to infinity while its 256^16.

Manu

4. Aug 6, 2004

### chroot

Staff Emeritus
As the number x in

$$\left(\frac{x-1}{x}\right)^{x}$$

approaches infinity, the result definitely does approach 1/e. Since 256^16 has plenty of significant digits, 1/e is close enough to an accuracy of only four decimals.

- Warren

5. Aug 6, 2004

### Gokul43201

Staff Emeritus
And that's what 0.3679 is !

6. Aug 6, 2004

### Manu2380

Hi Warren,

im such a dummy, lol, for 1/e in my windows calc. i kept using 2.178 which is answer a) for the constant e. Finally i looked e in my pocket handbook and found the right one, so it is answer c. And btw how did you use equation in the posts? Please do let me know tnx Warren.

Manu

7. Aug 6, 2004

### chroot

Staff Emeritus
The short answer is simply to click on the images to see their source code. You can just copy and paste the source into your own messages to include those equations. You'll figure out how it works in no time by example.

- Warren

8. Aug 6, 2004

### Gokul43201

Staff Emeritus
Manu, we have Chroot to thank for LaTex on PF.

9. Aug 7, 2004

### BobG

No one silly enough to unthinkingly punch this in on their calculator, huh? :tongue2:

10. Aug 8, 2004

### Gokul43201

Staff Emeritus
Strange I would have thought you'd get 1 :grumpy:

11. Aug 12, 2004

### AndrewEskeClarke

Would it not be d) 0.000 because in the parantheses would be a decimal and therefore as it grew exponentially it would lessen infinitely towards 0, and 0.000 just means that whatever number rounded to three decimal places would be 0? Or did y'all already establish that?

12. Aug 17, 2004

### Gaz031

>_< ...., whistles and looks over shoulder..

13. Aug 17, 2004

### Tau_Muon_PlanetEater

The answer is something like .9999999999999999999999999999999 but has many more nines than i can fit. So this rounds to 1.000, since the answers go to 3 decimals.

14. Aug 17, 2004

### Tau_Muon_PlanetEater

My mistake, the real answer is more like 0.000000000000000000000001 to many more decimal places, so it rounds to 0.00.