By Isaac Todhunter

This Elibron Classics booklet is a facsimile reprint of a 1864 version by way of Macmillan and Co., Cambridge and London.

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

**Sample text**

90. Use your calculator to evaluate 91. Use your calculator to evaluate 14489. Does the answer appear to be a rational or an irrational number? Why? 882 . Does the answer A 49 appear to be a rational or an irrational number? Why? 92. 2 Page 18 I N T E G E R E X P O N E N TS A N D S C I E N T I F I C N O TAT I O N S K I LLS O BJ E CTIVE S ■ ■ ■ C O N C E P TUAL O BJ E CTIVE S ■ Evaluate expressions involving integer exponents. Apply properties of exponents. Use scientiﬁc notation. ■ Visualize negative exponents as reciprocals.

27x3 y9 z6 c. qxd 22 8/7/12 5:34 PM Page 22 C H A P T E R 0 Prerequisites and Review Simplifying Exponential Expressions EXAMPLE 6 Write each expression so that all exponents are positive (assume all variables are nonzero). (x2 y-3) 2 a. (3x2 z-4) -3 b. -1 4 -3 (x y ) 3 c. (-2xy2) 3 2 -(6xz ) Solution (a): Apply the product to a power property. (3x2 z-4) -3 -3 -3 = (3) (x2) (z-4) Apply the power property. ϭ 3Ϫ3 xϪ6 z12 Apply the negative-integer exponent property. = Evaluate 33. = -3 z12 33 x6 z12 27x6 Solution (b): (x2 y-3) 2 Apply the product to a power property.

1 Real Numbers 11 Using the Distributive Property EXAMPLE 7 Use the distributive property to eliminate the parentheses. a. 3(x ϩ 5) b. 2(y Ϫ 6) Solution (a): ➤ 3(x ϩ 5) ϭ 3(x) ϩ 3(5) Use the distributive property. ϭ 3x ϩ 15 Perform the multiplication. Solution (b): ➤ 2(y Ϫ 6) ϭ 2(y) Ϫ 2(6) Use the distributive property. ϭ 2y Ϫ 12 Perform the multiplication. ■ YO U R T U R N ■ Use the distributive property to eliminate the parentheses. a. 2(x ϩ 3) Answer: a. 2x ϩ 6 b. 5y Ϫ 15 b. 5(y Ϫ 3) You also probably know the rules that apply when multiplying a negative real number.