53. Shew that (x2 + xy + y2) (a2 + ab +b3) can be expressed in the form X2 + XY + Y2. 54. Shew that (x2+pxy + qy3) (a2 + pab + qb3) can be expressed in the form X2+pXY+qY3. 55. Shew that, if 28 = a+b+c, (i) a (sb) (sc) + b (s −c) (s − a) + c (s—a) (s—b) (ii) (iii) = · (a + b) s ( s − c) + c ( s − a) ( s − b) — 2sab. (iv) a (b−c) (8 − a)2 + b (c − a) (s — b)2 + c (a - b) (8 − c )2 = 0. (v) 8 (8b) (8 c) + 8 (8 − c) (8 − a) + s (s − a) (s − b) -(8-a) (8-b) (sc)=abc. (vi) (s — a)2 (s — b)2 (s — c)2 + s2 (8 − b)2 (8 — c)2 + s ( s − a) ( s − b ) ( s − c) (a2 + b2 + c2) = a2b3c2. 56. Shew that, if 28= a+b+c+d, 4 (bc + ad)2 − (b2 + c2 — a2 — d2)2 = 16 ( s − a) ( s − b) (s – c) (8 — d). a (s-b) (s-c) (sd) + b ( s − c) (s − d) ( s − a) + c ( s − d) ( s − a) (8 — b) + d (s − a) (s - b) (s — c) + 2 ( s − a) (s —b) (s—c) (s — d) ―s (bcd+cda + dab + abc) · 57. == 2abcd. Shew that, if a+b+c+d= 0, then ad (a + d)2 + bc (a − d)2 + ab (a + b)2 + cd (a − b)2 + ac (a + c)2 + bd (a − c)2 + 4abcd = 0. 58. Shew that, if (a+b) (b+c) (c + d) (d+ a) then ac = bd. = (a+b+c+d) (bcd + cda + dab + abc) ; = 59. Shew that, if a+b+c=0 and x + y + z = 0, then 4 (ax+by+cz)3 − 3 (ax + by + cz) (a2 +b2 + c2) (x2 + y2 + z3) − 2 (b − c) (c − a) (a − b) (y − z) (z − x) (x − y) = 54abcxyz. 60. Shew that, if a + b + c = 0; then (i) 2 (a2 + b2+c2) = 7 abc (a* + b* + c*). (ii) 6 (a2 + b2 + c1) = 7 (a3 + b3 + c3) (a* +b*+c*). (iv) 25 (a2 + b2 + c2) (a3 + b3 + c3) = 21 (a3 + b3 + c3). (a3 + b3 + c3 + d3)2 = 9(bcd + cda + dab + abc)2 and none of the denominators be zero, then will l = m = n. 68. − a") (1 − a"~1) + ... (1 − a2)} + {(1 − a”) (1 − a” ̄1) ... (1 − a)} Shew that, if n be any positive integer 1-a" (1-a") (1 − a ̄1). (1 − a”) (1 — a′′−1) (1 − a′′−3) + 1 a 1- a2 + + 1- a3 (1 − a”) (1 − a”−1) ... (1 − a) 69. Prove that, if n be any positive integer, = n. + CHAPTER XIII. POWERS AND ROOTS. FRACTIONAL AND NEGATIVE INDICES. 158. The process by which the powers of quantities are obtained is often called involution; and the inverse process, namely that by which the roots of quantities are obtained, is called evolution. We proceed to consider some cases of involution, and of evolution. 159. Index Laws. We have proved in Art. 31, that when m and n are any positive integers, Thus the index of the product of any number of powers of the same quantity is the sum of the indices of the factors. Hence (am)n = amn ..(iii). Thus, to raise any power of a quantity to any other power, its original index must be multiplied by the index of the power to which it is to be raised. Again, to find (ab)m. (ab)"= = ab x ab x abx...... to m factors, by definition, = = (a×a× a...... to m factors) × (b × b × b...... to m factors), by the Commutative Law = am xbm, by definition. Hence Similarly (ab)m = am × bm. (abc.....)m = am × bm × cm ×... ...(iv). Thus, the mth power of a product is the product of the mth powers of its factors. The most general case of a monomial expression is a*b*c*...... Now (a*b1c*..............)TM = (a*)m (b2)m (c2).................from (iv) Thus any power of an expression is obtained by taking each of its factors to a power whose index is the product of its original index and the index of the power to which the whole expression is to be raised. As a particular case m m 1 am bm (1)TM = (a × } ) ̄ = a * × z — — — 7. 160. It follows from the Law of Signs that all powers of a positive quantity are positive, but that successive powers of a negative quantity are alternately positive and negative. For we have |