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Lesson 2

The Arithmetic of Powers

Multiplication by addition, and division by subtraction

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 2^{2} by another power of that same number, such as 2^{3}, you could find the product by expanding the powers and multiplying by ordinary arithmetic. Since 2^{2} = 4 and 2^{3} = 8, then (2^{2})(2^{3}) = (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 b^{n}. You will see shortly that this will lead to some interesting results.

Now, in the example (2^{2})(2^{3}) = (4)(8) = 32, we can also write 32 as a power of 2, because 32 = 2^{5}. So the entire equation can be written in powers.

(2^{2})(2^{3}) = 2^{5}

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 (3^{2})(3^{3}), would the product necessarily be some power of 3?