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### Comparing Numbers With Huge Exponents

```Date: 05/23/2002 at 16:12:24
From: Hunter Hill
Subject: Huge Exponents

Which is larger? 1999^1999 or 2000^1998

```

```
Date: 05/23/2002 at 17:49:54
From: Doctor Ian
Subject: Re: Huge Exponents

Hi Hunter,

Let's look at some smaller versions of the same problem:

n      n^n     (n+1)^(n-1)
-     -----    -----------
2         4              3
3        27             16
4       256            125
5      3125           1296
6     46656          16807

This is hardly a proof, but it suggests that the term with the
higher exponent will be larger.  If you can prove (e.g., by
induction) that n^n grows faster than (n+1)^(n-1), then it's got
to hold for the special case of n=1999.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 05/24/2002 at 15:19:56
From: Hunter Hill
Subject: Huge Exponents

Doctor Ian,

I appreciate the help. However, in order for me to get full credit
for this extra credit problem, I need to show my work and explain it
at a college algebra level. If possible, please assist. Thanks for

-Hunter Hill
```

```
Date: 05/24/2002 at 17:17:37
From: Doctor Ian
Subject: Re: Huge Exponents

Hi Hunter,

We're trying to show that n^n grows faster than (n+1)^(n-1).  One
way to do that is to show that the ratio

n^n
-----------
(n+1)^(n-1)

increases with increasing n, i.e., that if we choose some k to be
the value of n, then

k^k        (k+1)^(k+1)
----------- < -----------
(k+1)^(k-1)    (k+2)^k

^             ^
|             |
ratio for     ratio for
some k        next k

Since we know that all the terms are positive, we can
cross-multiply to get

k^k * (k+2)^k  <  (k+1)^(k+1) * (k+1)^(k-1)

If we can show that this inequality holds, then we've shown that
if k^k is larger than (k+1)^(k-1) for any any value of k, it
holds for every larger value of k.  And we've already shown that
it holds for k=2.

Since 1999 is larger than 2, it must be true for k=1999 as well.

Can you take it from here?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Exponents
High School Exponents

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