Multiplication of Powered Numbers

Date: 02/17/99 at 17:32:10
From: Duane Wilson
Subject: Multiplication of Powered Numbers

How do you figure out the product of two numbers with different powers, 
e.g. 20 to the power of 50 and 50 to the power of 20, and find out how 
many zeros are at the end of the number?

I have used my calculator and unfortunately the answer does not match 
any of the multiple choice answers. The number was too long. I've tried 
to add the results, to multiply without a calculator, to add the 
powers, and, finally, to add the integer and the powers.

Hope that you can help!

Date: 02/17/99 at 17:46:07
From: Doctor Pat
Subject: Re: Multiplication of Powered Numbers

Let's try a problem-solving trick. I call it "Start little and think 
big." Suppose the powers are easy:

    20^2 * 50^3  = 20*20*50*50*50 = 50000000
  
Why are there seven zeroes? Five of them are easy to find, so let's 
break those down some more.
     
       20*20*50*50*50 = 2*10*2*10*5*10*5*10*5*10
                      = 2*2*5*5*5*10*10*10*10*10
                      =... 

The five tens multiplied make five zeros on the end, but the other two 
come from two 2's multiplied by two 5's. So, if we had 20^a * 50^b,  
we would get a+b zeros from the tens, and we would get more zeros for 
every 2 * 5 that could be paired up, but this would just be the smaller 
of a or b. Let us test the theory:

20^6 * 50^5 on the calculator gives 2E16, which means 16 zeros after 
the 2 in front. 6 + 5 = 11, so eleven of them came from the tens, and 
since 5 is smaller than 6, we had five 2*5 pairs to make five more 
zeros, giving us 11. 
     
20^15 * 50^40 will have 40+15 zeros from the tens, and 15 from the 2*5 
pairs, giving 70 zeros in all. 

20^40* 50^30 will have 40 + 30 + 30 = 100 zeros at the end
   
The rule is exponents added plus one more of the smaller exponent. 
 
Good luck!

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