3 5. Required the difference of ✓ and V. Ans. #5V 15. 6. Required the difference of Vand V a Ans. 1 18. 7. Find the difference of ✓ 80a4x and 202*%3: Ans. 42"-24x XV 5%. 3 PROBLEM VI. To multiply surd quantities together. RULE. I. 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. 3 3 3 1. Required to find the product of 38 and 2/6. Here, 3X2X78X76=678X6=648=6N 16X3= 6X4XN3=24V3, the product required. 2. Required to find the product of V į and V. Here, XV XVSXVi;={XVEX X 815=*15=*Vis, the product required. 3. Required the product of 5V 8 and 3V 5. Ans. 30V 10. 4. Required the product of V6 and į V 18. Ans. 4. 3 3 5. Required 5. Required the product of Vs and Vio Ans. 1/35 6. Required the product of 18 and 5V 4. Ans. 10 V 9. 7. Required the product of aš and at. 3 Ans. a3]} 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 3 1. It is required to divide 8/108 by 22/6. 8;2X1086=4V18=479X2=4X3V2=1212 the quotient required. 2. It is required to divide 8V 512 by 412. 8-44=2, and 5121; 2}=256354x47; Therefore 2X4X 43=8x41=84, is the quotient required. 3. Let 6V/100 be divided by 32. Ans. 10/2. 4. Let 41000 be divided by 22 2V Ans. 10.02. 5. Let Vas be divided by V. Ans. V 3. 3 6. Let 3 Ans. IV 3. :: 6. Let V į be divided by v 7. Let Va, or fat, be divided by . Ans. opet PROBLEM VIII. To involve, or raise, surd quantities to any power. RULE. Multiply the index of the quantity by the index of tho 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 a First, T ={x}=;; 2/3 를 a=;Vi, the square required. 2. It is required to find the cube of V7 First, 7=x}x= Therefore evi'=7*=*#**343)*, the cuba required. 3. Required the square of 3V 3. . 4. Required the cube of 2, or v2. I 25 Ans. 2V 2. 5. Required 5. Required the 4th power of V0. 1 Ans. 273 Divide the index of the given quantity by the index of the root to be extracted ; then annex the result to the roog of the rational part, and it will give the root required. EXAMPLES. 3 1. It is required to find the square root of 9V 3. First, 1953 31 =3 xză is the square root required. Therefore 2. It * The square root of a binomial or residual surd, A+B, or 1 D 4-R, may be found thus : take vA-B=D; Then VA+B=v4+D+v A And vĀ–B=v4 D-4D. Thus, the square root of 8+2V7=i+v7; And the square root of 3-V8=v2-1; But for the cube, or any higher root, no general rule is given. 3 3. It is required to find the cube root of V 2, First, Vi=1; And 721}=27+3= 25, Therefore xV2;}=1 X2 is the cube root required. 3. Required the square root of 103. Ans. 10V 10. 4. Required the cube root of 3,0%. 5. Required the 4th root of 34*, Ans. 3*xxă 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. 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. EXAMPLES |