9. To add 6(a3 + a3¿2)3 and 7a(64a2 + 64b2) 3. 10. To find the sum of (a-m+5), 4(a−m+15) 3. 44 Ans. 16. 9 3 Ans. a 2a Ans. 34a(a2 + b2)3. -3(a-m+10)3, and Multiplication and division of surds. RULE. By Case II. reduce the surds to equivalent ones having the same index, and then multiply or divide as required. EXAMPLES. 1. To multiply 13 by 12, and divide the product by 5. Here the common index is 12, and the surds are equiva2, and 5; consequently, we shall have 12/36 × 24 12 lent to 3, √3 × 12 3 2. Find the product of 3a, 4a3, and 5a3. Ans. 60a Va. 6. Find the product of 7mbe, 5n Vef, and 4pve. Ans. 140mn3p 1⁄4bce3ƒ3. 7. Multiply 8a" by 4a", and divide 6a" by 3aa. 11. Multiply V + 2x3y + x2y2 by √x2 + xy. Ans. x(x+y). 12. Divide va3 + y3 by √x + y. Ans. Vx2 - xy + y2. CASE VI. To find powers and roots of surds. RULE. 1. To find a power of a surd, raise the surd to such a power as is denoted by the index of the power. 2. To find a root of a surd, extract such a root of the surd as is denoted by the index of the root to be extracted. EXAMPLES. 1. Find the square of 2 13 and the cube of 519. Here we have (213)2 = 2√3 × 2√3 = 22√3 = 4 x 3 = 12, and (59) = 593 125 x 9 = 1125. 2. To extract the square root of 9 1/2, and the cube root of 6417. Here we have (9 1/2) = 3√2, and (64√7)3 = 4√7. 3. Find the cube of abc. Ans. abe Ans. 243ab and ± 4c2ď3è113 ̧ 6. Find the cube root of $27ab3e12. Ans. Babe 8. Find the square of 3-2 and of a + √b. Ans. 11 6 1/2 and a2 + b + 2a vb. - 9. Find the square root of x + 2 Vxy + y and the cube root of a 3a2b + 3√ab2 -b. Ans. ±(√x + √y) and Va — Vb. 10. Find the square of 3 Va-4b and the square root of a + b + c + 2 Vab-2 Vac- 2 vbc. ±(√a Ans. 9a-24 Vab + 16b and ±(Va + √b — √c). CASE VII. To extract the square root of a binomial, supposing that one or both of its terms are under the sign of the square root. Assume = A + B and y = AB, then by addition. and multiplication we have 2 + y2 = A + B + A − B = 2A and a3y (A + B) × (A — B) = A2 - B2 or ay = √Aa — B2. From a2+ y2 = 2A and 2xy = 2 VA2-B, by addition and subtraction, we have x2+2xy + y2 = (x+y)2 = 2A+2√A2 — B2 consequently, by extracting the square roots and taking their sum and difference, we have = VA+WA2 — B2 + 2 The formulæ (a) and (b) will enable us to find the square roots of the binomials A + B and A - B, whether A and B are or are not affected by the sign of any root. EXAMPLES. 1. To extract the square root of 11 + 6 √2. Putting 11 for A and 6 1/2 for B, the binomial A + B becomes 11 + 6/2, as in the question; and we shall have √A2 - B2 = √112 — (6 1/2)2 of course from (a) we have required. 11+ 6 √2 = 3 + √2, as 2. To extract the square root of 8-21/15. Putting A8 and B = 215, the binomial A B be comes 8-215, as in the question; and we shall have WA2- B2 = 164-60= 2, and of course from (b) we get 1/8-2115=4/5 - √3, as required. 1/53, 3. To extract the square root of 5 + 12 √ — 1. Here we have A 5 and B 12 - 1, and A- B2 = = = √ 25 — (12 √ — 1)2 = 25+ 144 169 or VA-B213; con sequently, from (a) we get required. 5+ 12 √− 1 = 3 + 2 √-1, as 4. To extract the square root of √7 ± √3. 5. To extract the square root of a + b + 2 √ab. Ans. Va + √ō. 6. To extract the square root of 2a2· x2 + 2a √ a2 — x2. Ans. a + Va2 — x2. 7. To extract the square root of ±√1=0±√1. Putting 0 for A and V-1 for B, we get from (a) and ± 1± , as required. 8. Required, the square root of 3-4-1 and of -724 -1. Ans. 2-1 and 3-4-1. CASE VIII. To extract any root of a binomial, whether its terms are rational or surd. Assume (x + y)" = A + B and (xy)" = (A + B)(A — B) = A B C or C VA - B; conse quently, we shall have = ing the first member of the equation by the binomial theorem, n(n − 1) (n - 2) (n − 3), we have " + n(n - 1) 1 2 1 2. 3 4 -xr-bys +etc. A, and since (x + y)" × (x− y)" — (a2 — y2)" = A2 — B2 = C" gives a2—y2 = C or y2 = x2—C, the preceding equation is x-(x2-C)2 + etc. = A, (a'), y2 = x2 – C, (b'), which being solved give x and y; consequently, we easily get + y = A + B or x-y=VA-B, the nth root of A + B or A — B. 1. To extract the square root of 7 + 4 √3. = Here, by putting 7 for A, and 4 √3 for B, and 2 for n, we shall have A- B-17-16 x 3 = 1 C; consequently, (a) reduces to x2 + x2-1=2x2 - 1 = 7, or x2 = 4, or x=2, and from (b) ya C=4-1=3, or y = 13. Hence, we have 1/7 +4√3 = x + y = 2 + √3, as =2+13, as required. 2. To extract the cube root of 24. Put 24 48 24, A = 48, and B = 24, and we get VA-B12; consequently, since n = 3, (a') reduces to x2+3x(x2-12) 48, or a9a12, which is very nearly satisfied by putting a 3.5223. Putting the values of C and x in (b), we get ya — C = (3.5223)2 — 12 = 12.40659729 12 = 0.40659729, or y = 0.40659729 = 0.6376 very nearly. Hence, 48-24 = √24 = x − y = 3.5223 0.6376 = 2.8847, which is correct to three decimal places. |