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Additional resources for A Treatise On The Differential Calculus with numerous examples
What about the product? (e) Are the actions of washing laundry and drying it commutative? (f) Give an example of a pair of actions that is commutative. (g) Give an example of a pair of actions that is not commutative. 46. Commutative and Noncommutative Operations We have seen that addition and multiplication are both commutative operations. (a) Is subtraction commutative? (b) Is division of nonzero real numbers commutative? (c) Are the actions of putting on your socks and putting on your shoes commutative?
A ϭ amϩn 144 4244 43 1442443 m factors n factors 144424443 m ϩ n factors Thus aman ϭ amϩn. We would like this rule to be true even when m and n are 0 or negative integers. For instance, we must have 20 # 23 ϭ 20ϩ3 ϭ 23 But this can happen only if 20 ϭ 1. Likewise, we want to have 54 # 5Ϫ4 ϭ 54ϩ 1Ϫ42 ϭ 54Ϫ4 ϭ 50 ϭ 1 and this will be true if 5Ϫ4 ϭ 1/54. These observations lead to the following definition. ZERO AND NEGATIVE EXPONENTS If a 0 is any real number and n is a positive integer, then a0 ϭ 1 and a Ϫn ϭ 1 an Zero and Negative Exponents EXAMPLE 2 0 A 47 B ϭ1 1 1 (b) x Ϫ1 ϭ 1 ϭ x x (a) (c) 1Ϫ22 Ϫ3 ϭ 1 1 1 ϭϪ 3 ϭ Ϫ8 8 1Ϫ22 NOW TRY EXERCISE 19 ■ ▼ Rules for Working with Exponents Familiarity with the following rules is essential for our work with exponents and bases.
C) A drop of water contains more than 33 billion billion molecules. 90. (a) The distance from the earth to the sun is about 93 million miles. 000000000000000000000053 g. (c) The mass of the earth is about 5,970,000,000,000,000,000,000,000 kg. | Integer Exponents 21 S E C T I O N P. 3 91–96 ■ Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. who weighs W pounds and who is H inches tall, the bodymass index B is given by 91.