19. Simplify 3[1 − 2{ 1 − 4(1 − 3x)}], and find what quantity must be added to it to produce 3 - 8x.

20. Divide the sum of 10x2-7x(1+x2) and 3(x+x2+2) by 3(x2 + 1) − (x+1).

21. Simplify 5x1 – 8x3 – (2x2 -— 7) − (x2 + 5) + (3x3 − x), and subtract the result from 4x4 - x+2.

22. If a=0, b=1, c=3, d=-2, e=2, find the value of
(1) 3cbde; (2) (c+a)(c− a)+b2; (3) e +a3.
23. Find the product of 7x2 − y(x − 2y) und x(7x+y) − 2y2.
24. Subtract (a3+4)+(a2 − 2) from (a3+4)(a2 – 2).

25. Express by means of symbols

(1) b's excess over c is greater than a by 7.

(2) Three times the sum of a and 2b is less by 5 than the product of b and c.

26. Simplify

3a2 – (4a – b2) – { 2a2 — (3b − a2) – 2b – 3a} – { 5b -- 7a — (c2 — b2)}. 27. Find the continued product of

x2+xy + y2, x2 - xy + y2, x1- x2y2+y1.

28. Divide 4a2 -9b2-4ac + c2 by 2a-3b-c.

29. If a = 3, b=-2, c=0, d=2, find the value of
(1) c(a+b)+b(a+c) + a(c-b); (2) aa+da.

30. From a rod a+b inches long b-c inches are cut off; how much remains?

31. A boy buys a marbles, wins b, and loses c; how many has he then?

32. Simplify 2a - {5a-[Sa - (2b+a)]}, and find the value of

(a - b) [a2+b(a+b)] when a = 1, b=2.

33. Divide 1-5x+4x5 by x2-2x+1.

34. Multiply the sum of 3x2-5xy and 2xy - y2 by the excess of 3x2+y2 over 2y2+3xy.

35. Express in algebraical symbols

(1) Three times x diminished by the sum of y and twice z.

(2) Seven times a taken from three times b is equal to five times the product of c and d.

(3) The sum of m and n multiplied by their difference is equal to the difference of the squares of m and n.

36. If a = 2, b=1, c=0, d=-1, find the value of

(db)(c-b)+(ac − bd)2 + (c2 − d) (2c — b).

CHAPTER VIII.

REVISION OF ELEMENTARY RULES.

[If preferred, this chapter may be postponed until the chapters on Simple Equations and Problems have been read.]

Substitutions.

62. DEFINITION. The square root of any proposed expression is that quantity whose square, or second power, is equal to the given expression. Thus the square root of 81 is 9, because 92=81.

The square root of a is denoted by 2/a, or more simply a. Similarly the cube, fourth, fifth, &c., root of any expression is that quantity whose third, fourth, fifth, &c., power is equal to the given expression.

The roots are denoted by the symbols 3/, /, 5/, &c.

Examples. 3/27 = 3; because 33 = 27.

5/32 = 2; because 25 = 32.

The root symbol

is also called the radical sign.

Example 1. Find the value of 5√(6a3b1c), when a = 3, b= 1, c = 8. 5(Ga3b*c) = 5 × √(6 × 33 × 14 × 8)

Note. An expression of the form (6a3b+c) is often written √6abc, the line above being used as a vinculum indicating the square root of the expression taken as a whole.

Example 2. If a = 4, b = -3, c = −1, ƒ = 0, x = 4, find the value of 7(a cx)-3√b+c2+5√(fx).

The expression=7√(-4)-(-1)4-3 √( − 3) ( − 1)2 +0

= 7 3/( − 64) - 3√√S1
=7× (-4) - 3×9

=-55.

EXAMPLES VIII. a.

If a = 4, b=1, c=6, d=0, find the value of

If a=-3, b=2, c=-1, x = - 4, y=0, find the value of

11. ex. 12. √3ac3.

13. √6abx.

15. √3abcx. 16. V√a3c2.

17. b.

19. √3ac-√cx+√b°cx.

20. cy+2ab-√9a2.

21. If x=100, y = 81, z=16, find the value of

22. If a =-6, b=2, c=−1, x=-4, y=0, find the value of 2√a cx-2√a*b*xy+8a-b.

Fractional Coefficients and Indices.

63. Fractional Coefficients. The rules which have been already explained in the case of integral coefficients are still applicable when the coefficients are fractional.

Example 1. Find the sum of 3x2+xy-y2, - x2 - 3xy+2y2, 3x2-xy - žy2.

Example 2. Divide + y2+y3 by x+y.

x + y) { x2+xy2+ √by3 ( 4x2 - 1xy+1y2

64. Fractional Indices. In all the examples hitherto explained the indices have been integers, but expressions involving fractional and negative indices such as a3, x, 3x2+x3−2, a ̄2-4a ̄1x−3.2 may be dealt with by the same rules.

For

a complete discussion of the theory of Indices the student is referred to the Elementary Algebra, Chap. XXXI. It will be sufficient here to point out that the rules for combination of indices in multiplication and division given in Chapters v. and vi. are universally true.

3

Example 1. xxx = x2 + 1 = xH ̧

Example 2. a ̄1× a1 = a ̄ 4+ 4 = a° = 1. [See Note, Art. 50.]

Example 3.

2-31-1

Example 4. 3x2y1÷x3y1 = 3x2 -3 y1 ́1 = 3x ̄1y1.

-2

Example 5. a ̄2¿ 3÷a2b ̄ 2 - 2 b 3 + 1 = a + b

It will be seen from these illustrations that the rules for combining indices in multiplication and division may be concisely expressed by the two statements,

(1) am xan=am+n,

(2) am÷an=am-n;

where m and n may have any values positive or negative, integral or fractional.

65. We shall now give some examples involving compound expressions.

Example 1. Multiply 2-3x+4 by 2x-1.

Example 2. Multiply cx+2c-7 by 5-3c-x+2cx.

Example 3. Divide

3

5

24x1 – 16x ̄32+x1 – 16x ̄1– 5xa by 8x ̄1– 2x† +x1 – 4x1.

Arrange divisor and dividend in descending powers of x.

1. Find the sum of -m-n, -3m+n, −2m - n.

-

2. Add together ab+c, a − 1b, ja+11⁄2b+¦c, − ja+11⁄2b−jc.

3. From a+b-c take a−b+c.

4. Subtract a2+ab-1b2 from a2 - ab+b2.

5. Multiply x2+ y2 by x - y.

6. Find the product of

2-3x+1 and 1x+}.

7. Divide - y3 by x-by.

8.

Divide a3-2a2b+11ab2-363 by a2 - ab+b2.

9. Simplify (2x − 3y) − }(3x+2y) + 11⁄2 (7x − 5y). 10. Find the sum of

{y3 — — by2+ $y - }, }y2+} + {¡y, - {y2-ty+f2.

11. Find the product of

1 x − by + (z − y) and (x−2) – {(y− {x).

12. Simplify by removing brackets

8(-2)+5 (2a-3 (a-3)}.

13. Divide 3 − }}x2+}x − } by 3x-1.

14. Subtract (7x-9y) from }(x − 3y) – (y – 2x).

15. Add together (xy)(x+y) and (2x-ly)(x − y). 16. Multiply a3 - ‡a2x+4x3 by a -2x.

17. Divide 36a2+b2+1-4ab - 6a+3b by 6a-b-1. 18. Simplify 6{x − }(y −4)} {1(2x − y) +2(y − 1)}.