and the system of equations (B) will then become If in (D), we suppose n=5 and az=32, we shall have X1=81; X2=41; X3=21 ; X4=11; X5=6. The above supposition causes our question to agree with Ques. 13, Chap. XV, Higher Arithmetic. CHAPTER IV. INVOLUTION, EVOLUTION, IRRATIONAL AND IMAGINARY QUANTITIES. INVOLUTION. (94.) The process of raising a quantity to any proposed power is called INVOLUTION. When the quantity to be involved is a single letter, it is involved by placing the number denoting the power above it a little to the right. (Art. 11.) After the same manner we may represent the power of any quantity, by enclosing it within a parenthesis, and then treating it as a single letter. Thus, the second power of mx=(mx)”, = = &c. CASE I. (95.) To involve a monomial, 'we obviously have this RULE. I. Raise the coefficient to the required power, by actual multiplication. II. Raise the different letters to the required power by multiplying the exponents, which they already have, by the number denoting the power, observing that if no exponent is written, then one is always understood. To this power prefix the power of the coefficient. NOTE. If the quantity to be involved is negative, the signs of the even powers must be positive, and the signs of the odd powers negative. (Art. 30.) EXAMPLES. 1. What is the square of 3ax3 ? Here the square of 3 equals 32=3x3=9. Considering the exponent of a, in the expression ax3, as one, we find a2x for the square of ax3. 6 Therefore we have (3ax3)2=9a2x6. 2. What is the fifth power of -2ab3 ? Ans. (-2ab3)=-32a5b15. 1 3. What is the fourth power of —— xy-2? 2n 5. What is the third power of x®y-? Ans. xoy yo 6. What is the nth power of —2x-3y2 ? Ans. +Pox+3y2n =+2"yo 7. What is the square of –72-17-3 ? 49 Ans. 49x-2y-6= x?yo 1 8. What is the third power of - Sy xan -52 5 9 Ans. 195 *oy xy xz x9 125y15 9. What is the seventh power of -m*cz-2? Ans. -m*a?z-7. 10. What is the fourth power of -n-*y* ? . 2 2 3 16 Ans. 81 n-8y CASE II. (96.) When the quantity is compound, we can write the different powers by the aid of rules which we will hereafter point out. (See Binomial Theorem.) At present, we will content ourselves, by involving compound expressions, by actual multiplication according to Rule under Art. 34. |