Lesson 5.1 Exponents

Definition of Power: If b and n are counting numbers, except that b and n are not both zero, then exponentiation assigns to b and n a unique counting number b n, called a power. bnbn n is called the exponent. b is called the base. How do we compute exponents? The exponent counts the number of times that 1 is multiplied by the base. OR The exponent n counts the number of times that the expansion · b is joined to 1.

Power Form Number of Expansions Operator Model Basic Numeral · 3 · 3 · 3 · 381 four 3 1 · 3 · 3 · 327 three · 3 · 39 two · 33 one none In general, b n = 1 · b · b · b · … · b n times

For all b  0, b 0 = 1. Here are several more examples: 3 3 = 1 · 3 · 3 · 3 = = 1 · 3 · 3 = 93 1 = 1 · 3 = 33 0 = = 1 · 2 · 2 · 2 = 82 2 = 1 · 2 · 2 = 42 1 = 1 · 2 = 22 0 = = 1 · 1 · 1 · 1 = 11 2 = 1 · 1 · 1 = 11 1 = 1 · 1 = 11 0 = = 1 · 0 · 0 · 0 = 00 2 = 1 · 0 · 0 = 00 1 = 1 · 0 = 0 We have the following general rules: For all b, b 1 = b. For all n  0, 0 n = 0. ? ? What about 0 0 ? It is not defined.

Evaluate each power: By definition, 6 2 = 1 · 6 · 6 = 36 When the exponent is greater than 1, this can be shortened by dropping the 1 at the front and writing: 6 2 = 6 · 6 = We may compute this as 2 4 = 2 · 2 · 2 · 2 = Since the exponent is 0, we have to use the definition. 5 0 is 1 multiplied by 5 no times. That is, 5 0 = 1. We may use exponents to count the number of factors: 7 · 7 · 7 · 7 may be written as 7 4. b · b · b may be written as b 3. 5 · 3 · 3 · 5 · 3 · 3 may be written as 3 4 · 5 2.

Caution: You may have been told that 2 3 is 2 times itself 3 times. This is NOT TRUE !! We may write 2 3 = 2 · 2 · 2, but there are only 2 products here, not 3. Remember: For 2 3, the exponent 3 counts the number of times that 1 is multiplied by = 1 · 2 · 2 · 2