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### Multiplication of Powered Numbers

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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/
```
Associated Topics:
High School Exponents
Middle School Exponents

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