# ### Arithmetic of Computers

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### Lesson 2

#### Page 41

In Lesson 1 we learned how powers of numbers are written by using exponents. Exponents, of course, were designed to be more than decorative additions to our number system. They provide a quick and easy way to multiply numbers by adding, and to divide numbers by subtracting. This is essentially what engineers do when they use a “slide-rule”.
If you wanted to multiply one power of a number, such as 22 by another power of that same number, such as 23, you could find the product by expanding the powers and multiplying by ordinary arithmetic. Since 22 = 4 and 23 = 8, then (22)(23) = (4)(8) = 32. But when you are asked to find the product of two powers of the same number, the product also should be written as a power of the number. In other words, the answer will be of the form bn. You will see shortly that this will lead to some interesting results.
Now, in the example (22)(23) = (4)(8) = 32, we can also write 32 as a power of 2, because 32 = 25. So the entire equation can be written in powers.
(22)(23) = 25
In this case, the product is a power of the same base as the two numbers that were multiplied. If we multiplied powers of 3, say (32)(33), would the product necessarily be some power of 3?