(8) 4x1 + 4x2y2 — 12x2z2 + ya — 6y2x2 + 924. (9) 4x2 - 24xy + 36y2. (11) 9x1 — 24x3y + 16x2y2 + 6x2y — 8xy2 + y2. The pupil should compare the algebraical and arithmetical processes for finding the square root; he will find that tney are alike, and that the former is a proof of the latter. 1. The square of a fraction is found by squaring both numerator and denominator. 2. The square root of a fraction is found by taking the square root of both numerator and denominator. Similarly any power or root of a fraction is found by taking that power or root of both numerator and denominator. Since a × a = a2, ́and (—a) × (− a) = a2, it follows that a2 has two roots, a and -α. Thus the square root of any quantity when found may be either positive or negative. Exercise 28. MISCELLANEOUS EXAMPLES. (Selected from Government EXAMINATION PAPERS.) (1) Multiply 4a2 + 12ab + 9b2 by 4a2 — 12ab+962. The product of two factors is (3x+2y)3 −(2x + 3y)3, and one of the factors is x-y; find the other factor. (4a2 +962 +12ab) (4a2 + 962-12ab) = {(4a2 + 962) + 12ab} { (4a2 + 962) — 12ab} = 16a1+72a2b2 +81b1 — 144a2b2 =(27x3 +54x2y + 36xy2 + 8y3) — (8x3 + 36x3y + 54xy2 +27y3) =1923—1973 +18xy-18xy2 =19 (23-y3) + 18xy (xy) = 19 (x − y) (x2 + xy + y2) + 18xy (x −y) =(xy) {19 (x2 + xy + y2) + 18 xy} =(xy) (19 x2+37 xy +19 y2) .. the other factor is 19x2 + 37xy + 19y2. (2) Show that x (x + 1) (x + 2) (x + 3) + 1 = (x2 + 3x +1)2. (3) Find the value of x2 + (x2 − 42x + 89) when x = 2; also of _a +√ a2 + b2 a32b (a - b) when a = - b and b = -3. (4) (x+3)(x + 7) = (x − 3) (x -8). Find x. (5) Prove that (x2 + xy — y2)2 = (x2 — xy — y2)2+4xy (x2 —y2); and show that the equation holds good if x = 5 and y = 5. (6) (a) By how much is a greater than 50? (b) If I travel x miles at the rate of y miles an hour, what is the number of hours? (7) What are the rules for the removal of a bracket? (8) Find the value of (a + b) (b+c) (c + d) (d + a) − (b +c) (b − d). What is your result if b = 2, d = } ? (9) Simplify {x (x + 1)(x + 2) + x (x − 1) (x − 2)} + } (x − 1) x (x + 1). (10) Multiply b2 + 2ab —c2 +a2 by c2 + a2 — 2 ab + b2, and show that the result may be written in the form (a2 — b2)2 +c2 (4 ab - c2). (11) Add together 7x-11 + xyz, 16 −y + 8 √xyz; 5y-6x-10√xyz and 4 14y+9x 20; and subtract from the amount : 5x — 11y − 5% + 2√xyz — 16. (12) Simplify (a − b − c) + (b + c − d) — (c — d −ƒ) − (f + g− e). (14) Prove the rule for finding G. C. M. of two equations. (15) If a quantity C, be a common measure of A and B, it will also measure the sum or difference of any multiples of A and B as MA NB. (16) Show that a quantity may be transferred from one side of an equation to the other by changing the sign without destroying the equality expressed by it. (17) Simplify 3a — [a + b − {a + b + c − (a+b+c+d)}] and multiply the result by 2a + b + d. (18) (a + b)2 × (a — b)3. (19) Divide 2 + y2 + 1 − 2y + 2x- 2xy by x-y + 1, and find when x2 + ax + b is exactly divisible by x+y. (20) Find the continued product of a+b+c, − a + b + c, abc, and a + b — c. (21) Divide - 6x2 + 27x1 by } + 2x + 3x2. (22) What is an expression? When are terms said to be like? What is meant by the nth root of a quantity? Find the value of √2 −x + √2x-1-4r when r 12. |