cand add together its exponents, and the sum will be the ement of the product. 2. Multiply 3 a2 b by 6 a b3. 3. Multiply 8 a3 b2 x by 2 a2 b2 x. ANS. 18 a3 64. 4. Multiply 3 a5 b2 x y by 5 a b3 x3 y5. 7. Multiply 3 a2-5 a x3 by 2 a3 x2. 8. What is the product of a y— 6+ y3 multiplied by a3 y2? 9. Required the product of a3- 62 multiplied by a3 + b2. 10. Multiply a2 b c3 — 12 by 2 a2 b3. 11. Multiply x3 + x2 y + x y2 + y3 by x-y. 12. Required the product of a3 + a 6 multiplied by 2 a2+ a +1. 13. Multiply a2 + aa + a5 by a2 — 1. 14. Multiply 3 (x2 — y3)2 by a (x2 — y3)9. ANS. 3 α (x2 - y3)4. 15. Required the product of a2 (x3 12 + x2)3 multiplied by a b (x3 — 12 + x2)3. 16. Multiply 3 (a2 + b3)2 by 5 (a2 + b3)2. 17. Multiply a (x3 · y2 + 4)3 by 6 (x3 — y2 +4)3 18. Multiply 3 a2 (x3 — y3)2 by 4 a2 (x3 — y3)2. 19. Multiply 4 a2 — 16 a x + 3 x2 by 5 c3 —- 2 a2 x. 20. Multiply a— 2 a3 + 4 a2 b2 — 8 a b3 + b SECTION VII. 1. Divide a5 by a3 Division. ANS. a2. The exponent of the quotient added to that of the divisor, must be equal to the exponent of the dividend; for these two quantities are multiplied together by the adding of their exponents; and the product of the divisor and quotient must always give the dividend. In the above example, the dividend a5 = a a a a a, and the divisor a3 = a a a; and the division may be expressed thus, ααααα Now, if we cancel a a a in the numerator and denominator, there will be left in the dividend a a, that is, a2, as in the answer given above. Hence, if we are required to divide a power of any letter, as a5, by another power of the same letter, as a3, we must subtract the exponent of the divisor from the exponent of the dividend, and the remainder will be the exponent of the quotient. 2. Divide a by a3. 3. Divide a3 b4 by a b3. 4. Divide a b5 c2 by a6 b4 c. ANS. a3. 5. Divide 16 a1o b4 x2 by 4 a5 b3 x2. 6. Divide 39 a2 m3 y1 by 13 a2 m2 y3. 7. Divide a2 by a2. ANS. 1 By the rule above obtained, a2 ÷ a2 = ao; for a2 — 2 = ao. Cata2 = 1; for if we divide any quantity by itself, the quotient is 1. Hence we see, that a is always equal to 1, whatever may be the value of a; and the same is true of any other quantity which has for its exponent. 0 8. Divide a5 by a5. 9. Divide 5 a3 by 5 a3. 10. Divide a2 by a3. ANS. a-1. Here the exponent of the divisor is greater than that of the dividend; but the general rule must be observed. Thus, a2 ÷ a3 = a2 — 3 — a—1. In the expression a―1, a is said to have a negative exponent. So, also, a a2 = a−1 ; a ÷ a3 = a−2; a ÷ a1 = a3; a ÷ a3 — a -4; and so on. 11. Divide a by ao. 12. Divide a2 ba x3 by a5 b6 x7. 13. Divide ao by a. ANS. α-2. ANS. a-1. The division of ao by a, that is, by a1, may be thus expressed; ao÷a1 = ao-1 = a-1. But it will be have negative exponents. Thus, ao = 1; a-1= If the value of a be 2, then ao = 1; a−1 = 1; ANS. 1. 14. Divide (a + b)3 by (a + b)3. 15 Divide (a2 - x3 + b)4 by (a2 — ·x3 + b)2 16. Divide (a3 +5 − y)3 by (a3 + 5 — y)7. 17. Divide a b3 + 2 a3 b + 2 a2 b2 + a1 by a b2 + a3 + a2 b. Before we begin to divide compound quantities, we should arrange the terms of the divisor and dividend according to the powers of their letters, as this will greatly facilitate the work. The highest power of a letter should come first, and the lower powers should succeed in order. The first term of the divisor and the first term of the dividend should contain the same letter. To arrange this question, we place the letter of the divisor which is of the highest power, first, and the other terms in order, thus; a3 + a2 b + a b2. Now, as the first term of the divisor is a, the first term of the dividend should also contain a, and the whole should be arranged thus ; a++ 2 a3 b + 2 a2 b2 + a b3. a3 + a2 b + a b2 ) aa + 2 a3 b + 2 a2 b2 + a b3 ( a +b. a1+ a3b+ 20. Divide 3 +9 x2+4x-80 by x + 5. 21. Divide 68 16 c by b2 — 2 c2. 22. Divide a2 x — b2 x + 8 x − ̄`a2 y3 + b2 y3 — x+8x— 23. Divide ao - a1 x — a2 x3 + 2 x1 by a1 — x3. CHAPTER VIII. EQUATIONS OF THE FIRST DEGREE SECTION I. Introduction. WHEN two equal quantities, differently expressed, are compared together by means of the sign between them, such an expression is called an equation. Thus, 8+ 4 = 18 6 is an equation; for the sums are equal, though expressed in different numbers. So, too, if x5 and a 7 represent equal quantities, we have the equation x + 5 = a — 7. It is by means of equations that most of the investigations of Algebra are carried on; and the preceding chapters may be regarded as merely preparatory to this part of the science. An equation of the first degree contains only the first power of the unknown quantity, as x. When some higher power of the unknown quantity, as x2, or x3, enters into the equation, it is said to be of the second or third degree. The terms on the left of the sign, taken together, are called the first member of the equation; those on the right, the second member. Thus, in the equation |