By Arthur Cayley
This quantity is made out of electronic photos from the Cornell collage Library ancient arithmetic Monographs assortment.
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Using the method of trial and error, as in Example 8, we obtain 4x 2 Ϫ 12xy ϩ 9y 2 ϭ ͑2x Ϫ 3y͒ ͑2x Ϫ 3y͒ ϭ ͑2x Ϫ 3y͒2. Thus, 12x 2 Ϫ 36xy ϩ 27y 2 ϭ 3͑4x 2 Ϫ 12xy ϩ 9y 2͒ ϭ 3͑2x Ϫ 3y͒2. (b) Since each term has x 2y as a factor, we begin by writing 4x 4y Ϫ 11x 3y 2 ϩ 6x 2y 3 ϭ x 2y͑4x 2 Ϫ 11xy ϩ 6y 2͒. By trial and error, we obtain the factorization 4x 4y Ϫ 11x 3y 2 ϩ 6x 2y 3 ϭ x 2y͑4x Ϫ 3y͒͑x Ϫ 2y͒. L If a sum contains four or more terms, it may be possible to group the terms in a suitable manner and then find a factorization by using distributive properties.
ILLUS TRATION The Exponential Notation an 54 ϭ 5 и 5 и 5 и 5 ϭ 625 ͑ 12 ͒5 ϭ 12 и 12 и 12 и 12 и 12 ϭ 321 ͑Ϫ3͒3 ϭ ͑Ϫ3͒͑Ϫ3͒͑Ϫ3͒ ϭ Ϫ27 ͑Ϫ 31 ͒4 ϭ ͑Ϫ 31 ͒͑Ϫ 31 ͒͑Ϫ 31 ͒͑Ϫ 31 ͒ ϭ ͑ 19 ͒͑ 19 ͒ ϭ 811 It is important to note that if n is a positive integer, then an expression such as 3an means 3͑an͒, not ͑3a͒n. The real number 3 is the coefficient of an in the expression 3an. Similarly, Ϫ3an means ͑Ϫ3͒an, not ͑Ϫ3a͒n. ILLUS TRATION The Notation can 5 и 23 ϭ 5 и 8 ϭ 40 Ϫ5 и 23 ϭ Ϫ5 и 8 ϭ Ϫ40 Ϫ24 ϭ Ϫ͑24͒ ϭ Ϫ16 3͑Ϫ2͒3 ϭ 3͑Ϫ2͒͑Ϫ2͒͑Ϫ2͒ ϭ 3͑Ϫ8͒ ϭ Ϫ24 We next extend the definition of an to nonpositive exponents.
Approximate the realnumber expression to four decimal places. 08͒7/3 3/7 Exer. 93–94: Approximate the real-number expression to four decimal places. 9͒Ϫ2 (b) 527 95 Savings account One of the oldest banks in the United States is the Bank of America, founded in 1812. 04͒180 dollars. Approximate this amount to the nearest cent. 2 2h. Approximate the distance that can be seen from the top of the Chicago Sears Tower, which is 1454 feet tall. 46 2 W , where W is in kilograms and L is in meters. The largest documented halibut weighed 230 kilograms.
An elementary treatise on elliptic functions by Arthur Cayley