By Isaac Todhunter

This Elibron Classics e-book is a facsimile reprint of a 1864 version by means of Macmillan and Co., Cambridge and London.

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**Extra info for A Treatise On The Differential Calculus with numerous examples**

**Sample text**

3 Group like factors together. CONCEPTUAL In Exercises 77–80, determine whether each of the following statements is true or false. 77. Ϫ2 ϭ (Ϫ2) , if n is an integer. n n 83. Ϫa2 ϩ 2ab for a ϭ Ϫ2, b ϭ 3 84. 2a3 Ϫ 7a2 for a ϭ 4 78. Any nonzero real number raised to the zero power is one. n+1 79. In Exercises 83–86, evaluate the expression for the given value. x = x for x ϭ any real number. xn 85. Ϫ16t2 ϩ 100t for t ϭ 3 86. 80. xϪ1 ϩ xϪ2 ϭ xϪ3 a3 - 27 for a ϭ Ϫ2 a - 4 n k 81. Simplify A (am) B .

3 (x2) = x8 (x2) = x6 (2x) ϭ 2x 3 C ORRECT 3 2 # 24 ϭ 47 23 # 35 ϭ 68 3 Exponents should be multiplied (not raised to a power). (2x) ϭ 8x 3 3 Both factors (the 2 and the x) should be cubed. 2 # 24 ϭ 27 23 # 35 3 The original common base should be retained. The properties of integer exponents require the same base. We will now use properties of integer exponents to simplify exponential expressions. An exponential expression is simpliﬁed when: ■ ■ ■ ■ All parentheses (groupings) have been eliminated.

16t2 ϩ 100t for t ϭ 3 86. 80. xϪ1 ϩ xϪ2 ϭ xϪ3 a3 - 27 for a ϭ Ϫ2 a - 4 n k 81. Simplify A (am) B . -n -k 82. Simplify A(a-m) B . ■ CHALLENGE 87. 5 ϫ 108 square kilometers of land on the surface of the Earth. If one square kilometer is approximately 247 acres, how many acres per person are there on Earth? Round to the nearest tenth of an acre. 89. Evaluate: (4 * 10-23)(3 * 1012) -10 . Express your answer in (6 * 10 ) both scientiﬁc and decimal notation. ■ 88. 79 ϫ 106 square miles of land in the United States.