1 PROBLEM VI. To multiply surd quantities together. RULE. r. Reduce the surds to the same index. 2. Multiply the rational quantities together, and the surds together. 3. Then the latter product, annexed to the former, will give the whole product required; which must be reduced to its most simple terms. EXAMPLES. 1. Required to find the product of 38 and 26. Here, 3×2×8×6=6√8×6=6/48=6/16×3= 6X4X3 24/3, the product required. 3 3 3 √15=1515, the product required. 3. Required the product of 5/8 and 35. Ans. 30V 10. 4. Required the product of 16 and √ 18. 3 Ans. √4. 5. Required 3 3 6. Required the product of ✓ 18 and 54. 3 Ans. 109. 7. Required the product of af and a3. Ans. a313 or a. PROBLEM VII. To divide one surd quantity by another. RULE. 1. Reduce the surds to the same index. 2. Then take the quotient of the rational quantities, and to it annex the quotient of the surds, and it will give the whole quotient required; which must be reduced to its most simple terms. EXAMPLES. 1. It is required to divide 8 108 by 2/6. 82×108÷6=418=49×2=4X3√2=12√2 the quotient required. 3 3 2. It is required to divide 8 512 by 4/2. I I I I 8÷4=2, and 512525 = 2563 =4×43; Therefore 2×4×48×4=84 4, is the quotient re To involve, or raise, surd quantities to any power. RULE. Multiply the index of the quantity by the index of the power, to which it is to be raised; and annex 'the result to the power of the rational parts, and it will give the power required. EXAMPLES. 2 1. It is required to find the square of a 2. It is required to find the cube of 7 5. Required the 4th power of 6. 6. It is required to find the nth power of a' am Divide the index of the given quantity by the index of the root to be extracted; then annex the result to the root of the rational part, and it will give the root required. EXAMPLES. 3 1. It is required to find the square root of 93. First, √9=3; I And 3323 I 3 12 I Therefore 9/3 =3X3 is the square root required. 2. It * The square root of a binomial or residual surd, A+B, or A-B, may be found thus: take √A-BD; Then √A+B=VA+D+VAD, 1 2 2 Thus, the square root of 8+2√7=1+√7; But for the cube, or any higher root, no general rule is given. : 2. It is required to find the cube root of 2. Therefore 8 2X2 28 is the cube root required. 3. Required the square root of 103. Ans. 10V10. Ans. a I Ans. 34Xx INFINITE SERIES, AN INFINITE SERIES is formed from a fraction, having a compound denominator, or by extracting the root of a surd quantity; and is such as, being continued, would run on infinitely, in the manner of some decimal fractions. But by obtaining a few of the first terms, the law of the progression will be manifest, so that the series may be continued without the continuance of the operation, by which the first terms are found. Divide the numerator by the denominator; and the operation, continued as far as may be thought necessary, will give the series required. EXAMPLES. |