PROBLEM IV. To add surd quantities together. RULE. 1. Reduce such quantities, as have unlike indices, to other equivalent ones, having a common index. 2. Reduce the fractions to a common denominator, and the quantities to their most simple terms. 3. Then, if the surd part be the same in all of them, 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 quanti ties, they can only be added by the signs EXAMPLES. and 1. It is required to add 27 and 48 together. First, √279×3=3√3 ; Then, 3√3+4√3=3+4×√3=7√3= sum required. 3 2. It is required to add √500 and 108 together. Then, 54+34=5+3XV4=8V4 sum required. 3. Required the sum of 72 and 128. To subtract, or find the difference of, surd quantities. RULE. Prepare the quantities as for addition, and the difference of the rational parts, annexed to the common surd, will give 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. Required to find the difference of 448 and 112. Then 8/7-4747 the difference required. 2. Required to find the difference of 1923 and 243. First, 1923 =64×313 = 4×33; And 243= 8X3133⁄4=2X33; Then, 4×35—2×33=2X3 the difference re 7. Find the difference of ✔80a*x and √20a* x3. Ans. 40-24x X✓5x. PROBLEM VI. To multiply surd quantities together. RULE. 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 3/8 and 26. Here, 3X2X8X√6=6√8×6—6√48=6√16X3= 6X4X324/3, the product required. 3 2. Required to find the product of √ and 2. 151515, the product required. 3. Required the product of 5√ 8 and 3√ 5. Ans. 301 10. Ans. √4. 4. Required the product of √6 and √ 18. 5. Required 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 24/6. 82X/1086=4√18=4√9X2=4X3√2=12/2 the quotient required. 3 2. It is required to divide 8512 by 4/2. 8÷4=2, and 5123⁄4÷23=2563=4×43; I 3 Therefore 2X4X 48×48/4, is the quotient required. 3. Let 6/100 be divided by 32. Ans. 10/2. 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 a 2 3 Therefore a2 =÷•al3=÷Va2, the square re quired. 2. It is required to find the cube of √7. First, |=4×÷×÷=}}{{}; 3 I Therefore &√7] =÷÷÷7=343], the cube 독기 |