And the same method may easily be applied to examples with three or more surds. PROBLEM IV. To add Surd Quantities together 1. BRING all fractions to a common denominator, and reduce the quantities to their simplest terms, as in the last problem.-2. Reduce also such quantities as have unlike indices to other equivalent ones, having a common index.3. Then if the surd part be the same in them all, annex it to the sum of the rational parts, with the sign of multiplication, and it will give the total sum required. But if the surd part be not the same in all the quantities, they can only be added by the signs + and -. EXAMPLES. 1. Required to add 18 and 32 together. Ans. 73. First,/18 (9×2) = 3/2; and 32 (16×2)=4/2: Then, 3/2+4/2=(3+4)√/2=72= sum required. 2. It is required to add 3/375, and 3/192 together. First,3/375=3/(125×3)=53/3;and3/192=3/(64×3)=43/3: Then, 5/3+ 4/3 = (5+4)/39/3 = sum required. 3. Required the sum of/27 and /48. 4. Required the sum of 50 and √72. 5. Required the sum of and ✔i. 6. Required the sum of 3/56 and 3/189. 7. Required the sum of and 3/32 8. Required the sum of 3/ab and 5✓16a1b. Ans. 11/2. Ans. 15. Ans. 53/7. PROBLEM V. To find the Difference of Surd Quantities. PREPARE the quantities the same way as in the last rule; then subtract the rational parts, and to the remainder annex the common-surd, for the difference of the surds required. But if the quantities have no common surd, they can only be subtracted by means of the sign EXAMPLES. 1. To find the difference between ✔320 and 80. First,/320=√(64×5)=8✓5;and/80/(16×5)=45. = Then, 8/5 4/5 4/5 the difference sought. 2. To find the difference between 2/128 and 3/54. First,/128/(64×2)=4/2; and 3/543/(27×2)=33/2. Then, 4/233/23/2, the difference required. 3. Required the difference of 75 and 48. 4. Required the difference of 3/256 and /32. 5. Required the difference of and 6. Find the difference of ✔ and ✓. Ans. 3. Ans 22/4. Ans. √3. Ans. 6. 7. Required the difference of / and /. Ans./75. 8. Find the difference of 24a2b3 and √54*. PROBLEM VI. Ans. (362-2ab)/6. To multiply Surd Quantities together. REDUCE the surds to the same index, if necessary; next multiply the rational quantities together, and the surds together; then annex the one product to the other for the whole product required; which may be reduced to more simple terms if necessary. EXAMPLES. 1. Required to find the product of 4/12 and 3/2. Here, 4×3× √12×√✓/2=12√✓✓/(12×2)=12✓✓/24=12/(4x6) =12×2×6=24/6, the product required. 2. Required to multiply by V. Here = 32 === XX18 18, the product required. 3. Required the product of 3/2 and 28. Ans. 24. 4. Required the product of 1/4 and 3/12. Ans. 11/6. 5. To find the product of✓ and 3. Ans. 15. 6. Required the product of 22/14 and 33/4. Ans. 123/7. 7. Required the product of 2a3 and a3. Ans. 2a2. 8. Required the product of (a+b) and (a+b)a. 9. Required the product of 2x+b and 2x - b. vb. 10. Required the product of (a+26), and (a−2,/b)‡. 11. Required the product of 24 and 34. 12. Required the product of 4x4 and 2y. PROBLEM VII. To divide one Surd Quantity by another. REDUCE the surds to the same index, if necessary; then take the quotient of the rational quantities, and annex it to the quotient of the surds, and it will give the whole quotient required; which may be reduced to more simple terms if requisite. EXAMPLES. 1. Required to divide 696 by 3/8. Here 6÷3.√(96 ÷ 8) = 2/12 = 2 √ (4×3) = 2 × 24/3 4/3, the quotient required. 2. Required to divide 123/280 by 33/5. Here 12 ÷ 3 = 4, and 280 ÷ 5 = 56 = 8 × 7 = 23.7 ; Therefore 4 × 2 ×3/7 = 83/7, is the quotient required. 3. Let 4/50 be divided by 25. 4. Let 63/100 be divided 33/5. Ans. 2/10. Ans. 23/20. To involve or raise Surd Quantities to any Power. RAISE both the rational part and the surd part. Or multitiply the index of the quantity by the index of the power to which it is to be raised, and to the result annex the power of the rational parts, which will give the power required. EXAMPLES. 1. Required to find the square of şaa. Firs', (?)2 = 3 × 3 = %, and (a) = a1 × 2 = a2 = a. 2. Required to find the square of ļa3. First, × ↓ = ¦, and (a3)2 = at = a/a; Therefore ({a3)2 = a/a is the square required. 3. Required to find the cube of 6 or × 6$. First, () = 3 × 3 × } = 1⁄2, and (64)3 = 6} = 6 √6 ; Theref. (6)3 = × 6v6 = 166, the cube required. 4. Required the square of 23/2. 5. Required the cube of 32, or √3. 6. Required the 3d power of¦ √3. 7. Required to find the 4th power of }✔√2. I 8. Required to find the mth power of a". 9. Required to find the square of 2 +3. Ans. 4/4. Ans. 3√3. Ans. v 3. PROBLEM IX. To evolve or extract the Roots of Surd Quantities*. EXTRACT both the rational part and the surd part. Or divide the index of the given quantity by the index of the * The square root of a binomial or residual surd, a + b, or a b may be found thus: Take va2 — b2 = c ; Thus, the square root of 4+ 2√3 = 1 + √3; and the square root of 6—21/5 = √5 — 1. - But for the cnbe, or any higher root, no general rule is known. For more on the subject of Surds, see Bonrycastle's Algebra, the 8vo. edition, and the Elementary Treatise of Abgebra, by Mr. J. R. Young. root to be extracted; then to the result annex the root of the rational part, which will give the root required. EXAMPLES. 1. Required to find the square root of 16/6. 2 First, ✓16=4, and (61)2 = 6a ÷2 = 61; : theref. (166) = 4.644/6, is the sq. root required. 2. Required to find the cube root of 24 √3. 1 First, /= }, and (√3)3 = 3!÷3 = 3* ; =3 theref. (√3,=. .35 =1/3, is the cube root required. 3. Required the square root of 63. 4. Required the cube root of a3b. 5. Required the 4th root of 16a2. Ans. 6/6. Ans. 1a/b. Ans. 2/a. 6. Required to find the mth root of 7. Required the square root of a3-6ab+96. ARITHMETICAL PROPORTION AND PRO. ARITHMETICAL PROPORTION is the relation which two quantities, of the same kind, bear to each other, in respect to their difference. Four quantities are said to be in Arithmetical Proportion, when the difference between the first and second is equ... to the difference between the third and fourth. Thus, 3, 7, 12, 16, and a, a + b, c, c + b, are arithmetically proportional. Arithmetical Progression is when a series of quantities either increase or decrease by the same common difference. Thus, 1, 3, 5, 7, 9, 11, &c. and a, a + b, a + 26, a +36, a + 4b, a + 5b, &c. are series in arithmetical progression, whose common differences are 2 and b. The most useful part of arithmetical proportion and progression has been exhibited in the Arithmetic. The same may be given algebraically, thus: |