EXAMPLES. 1. Required to find the product of 4✓ 12, and 3✓ 2. Here, 4 × 3 × √12× √✓/?=12/12×2=12/24=124X6 = 12 × 2 × √6 246, the product required. 2. Require to multiply by } // }. 32 Here XXV ===÷18 18, the product required. = 3. Required the product of 32 and 28. 4. Required the product of 5. To find the product of 4 and 33/12. and ✔. 6. Required the product of 22/14 and 33/4. 7. Required the product of 2a3 and at. Ans. 24. A 1/6. Ans./15. Ans. 122/7. Ans. 2a2. 8. Required the product of (a + b) and (a + b)a3. I I and 3xTM. I I 12. Required the product of 4 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 6 96 by 3/8. Here 6÷3.✓ (96 ÷ 8) = 2√✓✓/12=2√✓✓(4×3)=2×2√3. = 4√3, the quotient required. 2. Required to divide 123/280 by 33/5. Here 123 = 4, and 280 ÷ 5 = 56 = 8X7 23 7 ; Therefore 4 × 2 × 3/7 = 83/7, is the quotient required. To Involve or Raise Surd Quantities to any Power. RAISE both the rational part and the surd part. Or mul tiply 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 a. First, (3)2 = 3 × 3 = ‚%, and (a1)2 = a‡ × 2 = a} 9 169 Therefore (a) =a, is the square required. 2. Required to find the square of ža3 First, X = and (a3)2 2 = a = a/a ; Therefore (a) = a/a is the square required. 3 = a. 1o, and (64)3 = 6a = 6 √/ 6. 27 First, (3) X X = 4. Required the square of 2 3/2. 1 8. Required to find the mth power of a”. 9. Required to find the square of 2 +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 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 First, ✔✅ 16 = 4, and (61) theref. (166) = 4. square root of 166. 6+ 4/6, is the sq. root required. 2. Required to find the cube root of✔3. First, √ =, and (✓ 3) = 3 ÷ 3 = 38; theref (3). 35/3, is the cube root required. 3. Required the square root of 63. 4. Required the cube root of a3b. Ans. 66. Ans. 1a / 6. >> The square root of a binomial or residual surd, a + b, or a b may be found thus: Take a+c = c; Thus the square root of 4+2 √3 = 1 + ✔√3; and the square root of 6 2√5 √5 But for the cube, or any higher root, no general rule is known. INFINITE INFINITE SERIES. AN Infinite Series is formed either from division, dividing by a compound divisor, or by extracting the root of a compound surd quantity; and is such as, being continued, would run on infinitely, in the manner of a continued decimal fraction. But, by obtaining a few of the first terms, the law of the progression will be manifest; so that the series may thence be continued, without actually performing the whole operation. PROBLEM I To Reduce Fractional Quantities into Infinite Series by Division. DIVIDE the numerator by the denominator, as in common division; then the operation, continued as far as may be thought necessary, will give the infinite series required. To Reduce a Compound Surd into an Infinite Series. EXTRACT the root as in common arithmetic; then the operation, continued as far as may be thought necessary, will give the series required. But this method is chiefly of use in extracting the square root, the operation being too tedious for the higher powers. EXAMPLES. |