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Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
You are correct. If we multiply two powers of the same base, we will get a product which is a power of that same base. So we can say that if 3m and 3n are two powers of 3, with exponents m and n, then their product is also some power of 3 - with a different exponent, of course. So our problem boils down to simply finding the exponent of the product.
Now, we know the base will be 3, so
(3m)(3n) = 3(exponentoftheproduct)
We do not know as yet what the exponent of the product is.
The rule for finding this unknown exponent is very simple:
When two numbers that are powers of the same base are multiplied, the exponent of the product is the sum of the exponents of the two numbers multiplied; that is,
(3m)(3n) = 3(m + n)
and in general (bm)(bn) = b(m + n).
Apply this rule to the multiplication of 42 by 41. What is the product of (42)(41), expressed as a power of 4?
