EXAMPLES. 1. Required to find the difference of 448 and V112. 448-647-87; First, And √112 V16x7=4√7; Then 87-474/7 the difference required. 2. Required to find the difference of 1923 and 24. First, 1923 = 64×3|32 = 4×33; And 243=8x3|3=2×5}; Then, 4x32x32x33 the difference required. 3. Required the difference of 250 and 18. 9 Ans. 7V2. Ans. 2x51. Ans./15. 6. Required the difference of 3 and 3." 1. 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=648=6√16x3=6x4x 3=243, the product required. 3 2. Required to find the product of ✓ and 3. Here, ×3 3/{=}×3/}}={x®√}}={x}×3√15 }3√15=}3√15, the product required. 3. Required the product of 58 and 35. Ans. 30/10. 4. Required the product of 36 and 3/18. 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 8108 by 26. 8÷2×✔108÷÷6=4√18=4√9×2=4×3√2=12√2 the quo tient required. 2. It is required to divide 83✓512 by 43√2. 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. 1. It is required to find the square of a3. Therefore :| =÷.a3=;3 3, the square required. 2. It is required to find the cube of $✔7. First, *=4×4×4=3?{}; FF; 3 Therefore 1.744.343, the cube required. 3, Required the square of 33, Arfs. 9°✔9. 4. Required the cube of 2, or √2. 5. Required the 4th power of 6. Ans. 2/2. 6. It is required to find the nth power of a. Ans. d PROBLEM IX. To extract the roots of surd quantities. RULE.* 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. 1. It is required to find the square root of 933. Therefore 933x33 is the square root required. 2. It is required to find the cube root of √2. First; The square root of a binomial or residual surd, A+B, or A-B, may be found thus: take ✔ —B2=D ; Thus, the square root of 8+27=1+√7;. And the square root of 3-8=√2—1; But for the cube, or any higher root, no general rule is given. Therefore 2×2 is the cube root required. 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. PROBLEM I. To reduce fractional quantities to infinite series. RULE. Divide the numerator by the denominator; and the operation, continued as far as may be thought necessary, will give the series required. 1−x) 1 (1+x+x® +x3+x®+, &c.=; and is the an swer. |