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### Smaller vs. Larger Exponents and Numbers

```Date: 03/24/2003 at 12:54:07
From: Toni
Subject: Numbers and exponents

Is a smaller number with a large exponent greater than a larger
number with a smaller exponent?

Using the numbers 1,2,3,4 what is the largest number you could make?
I'm thinking of 2^431 or 21^43.
```

```
Date: 03/25/2003 at 15:10:36
From: Doctor Ian
Subject: Re: Numbers and exponents

Hi Toni,

Generally, it's big exponents that give you power. Compare

99
2

to

2
99

However, the 'logic' you use to answer a question like this involves
understanding how exponents work, which is explained here:

Properties of Exponents
http://mathforum.org/library/drmath/view/57293.html

So before reading on, make sure you understand the basic rules.

431         43
2      ?   21

one thing you can do is try to get similar exponents. For example,

431          10 43
2     = 2 * (2  )

43
= 2 * (1024)

Now, 1024 raised to the 43rd power is going to be a _lot_ bigger than
21 raised to the same power, right?

Another way to do the comparison is to try to get close to powers of
10. For example, since 2^10 is close to 1000, we can approximate

431     10 43     ('~' means 'approximately equals')
2    ~ (2  )

3 43
~ (10 )

129
~ 10

Let's see what happens with 21^43:

43         43
21   ~ (2*10)

43     43
~ 2   * 10          Now we can use the same trick.

10 4     43
~ (2  )  * 10

3 4     43
~ (10 )  * 10

12     43
~ 10   * 10

55
~ 10

or anything else.

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

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