x2 324 C 16y? 8.x Example. Find the square root of 24+ x2 y Arrange the expression in descending powers of y. 16y: 327 8.0 x2 4y 4+ 32y +24 C + y 8.0 y yo Here the second term in the root, - 4, arises from division of By 8y and the third term, arises from division of 8 by ; 2 y' thus 8; -8x EXAMPLES XXIII. C. 32y by 2xy x2 + 22 +4. 323 9.x2 Find the square root of each of the following expressions: x2 4.xC x2 1. - 3x +9. 2. 4 + 4 y X2 10x . + +ya 4. +25. 25 5 ข X2 2.x 2ax a? 5. 6. + 4y2 y y2 by 72• 64.72 25 7. + +4. 8. - 2 + 25 1 9. + 10. 24+2.203 29 11. 1 - 3a3+ a?. 12. 24 – 2x + + 9 22 – 6.03. g 3 a4 322 1 14. 24 - 2.33+ + 169 ar2 а2 4ax 15. +4x2+ - 2.23 3 9 To FIND THE CUBE Root OF A COMPOUND EXPRESSION. 183. Since the cube of a +b is a3 + 3a-b+ 3ab2 +68, we have to discover a process by which a and b, the terms of the root, can be found when a3 +3a2b+3ab2 +63 is given. The first term a is the cube root of a?. Arrange the terms according to powers of one letter a; then the first term is a, and its cube root a. Set this down as the first term of the required root. Subtract a3 from the given expression and the remainder is 3a6 +3ab2 + b3 or (3a + 3ab+62) xb. Thus b, the second term of the root, will be the quotient when the remainder is divided by 3a2 +3ab+62. This divisor consists of three terms: 1. Three times the square of a, the term of the root already found. 2. Three times the product of this first term a, and the new term b. 3. The square of b. a +3aPb+3al2 +13 (a+b a3 3a25+3ab2 +33 3a2+3ab+12 3a2b+ 3ab2 +63 8.x3 – 36x2y + 54xya – 27y3 (2x – 3y 8.03 3 (2x) =12.x? - 36x2y + 54xy – 2743 +9yo +3ab +62 Example 2. Find the cube root of 27 +108x + 90x2 – 80.43 – 60x4 +48x5 - 8x6. 27 +108x + 90x2 – 80.x3 – 60x4 + 48.05 – 8.36 (3 + 4x – 2x2 27 108.2 + 90x" – 80x3 +16.x2 27 +36x + 16.x2 108.2 + 144.x2 + 64.23 54.x” – 144.23 – 60.x4 + 48.25 – 8.26 + 4x4 54x2 – 144.x3 – 60.x4 + 48x5 – 826 Explanation. When we have obtained two terms in the root, 3+4x, we have a remainder - 54x2 - 144x3 – 60x4 +48.05 – 8x6. EXAMPLES XXIII. d. Find the cube root of each of the following expressions : 1. Q3 +3a2+3a +1. 2. 23+ 6x2 + 12x +8. 3. a°23 – 3a_x_y2 + 3axy4 – yo. 4. 8m3 – 12m2 +6m – 1. 5. 6493 – 144a2b+108ab2 – 2763. 6. 1+ 3x + 6x2 +7.203 +6x4 +3.25 + 206. 7. 1-6x+21x2 – 44.23 +63x4 – 54.25 +27x6. 8. 43 +6a26 – 3a c+12ab2 – 12abc+ 3ac2 +863 – 126*c+66c2 – 63. 9. 846 – 365 +66a4 – 63a’ +33a? – Sa+1. 10. y - 3y +6y4-773 +672 - 3y +1. 11. 826 + 12.25 – 30.x4 - 35x3+45x2 +27x – 27. 12. 276 – 54.coa+117cao – 116ca3+1172*a* – 54c5+27a8. 13. 2726 – 27.26 – 1824 +17.003 + 6x2 – 3x – 1. 14. 24.x+y2 +96cRyt - 6.06y + 26 - 96xy6 +6476 - 56.2-y3. 15. 216+342x2 + 171x4 +27x6 – 27.25 – 109.x3 – 108x. 184. We add some examples of cube root where fractional terms occur in the given expressions. Example. Find the cube root of 54 – 27x3 + 8 36 X6x3 Arrange the expression in ascending powers of x. 8 36 +54 – 27x3 3.0 yo EXAMPLES XXIII. e. 23 1. 222 + -1. 2. + +4x+8. 8 4 2 27 3 2 27.23 27x2 93 3. 843 – 4.x?y2 +õryt 4. + 8. 27' 64y3 Hy2 + y 27 27 206 5. 23 - 9x + 6. -6.74 +12.cy3 - 8y. 2003 9x 7. 9y, 6y2 18 27 27 8. + 2x – 7+ 1080 48a2 8a3 9. + - 112 + X x3 6x2 33 33 + + 23 3 x2 203 + -7+ ta a 27+ 66 βα a3 3a? 372 73 11. + a b 13 79 60.x4 80.23 90x2 8x6 108x 48.25 12. + + yo у° 185. The fourth root of an expression is obtained by extracting the square root of the square root of the expression. Similarly by successive applications of the rule for finding the square root, we may find the eighth, sixteenth...root. The sixth root of an expression is found by taking the cube root of the square root, or the square root of the cube root. Similarly by combining the two processes for extraction of cube and square roots, certain other higher roots may be obtained. Example 1. Find the fourth root of 81x4 - 216x’y + 216.xoy- 96xy3 +1644. Extracting the square root by the rule we obtain 9x2 - 12xy + 4yo; and by inspection, the square root of this is 3x – 2y, which is the required fourth root. |