tity must be first raised to the power denoted by the sign, and then multiplied into the number or quan. tity under the radical sign. Thus, in the expression 3/7, if it be required to multiply the 3, into the surd part 7, the 3 must be squared (as denoted by the sign) which will be 9, then reducing this 9 to its original value, 3, it becomes 9, therefore 9x7=√/63. 3 Also, in the expression 3ax/2xy, the 3ax must be cubed, (as denoted by the radical sign) and it becomes 27a3x3, therefore as above, 27a3x3 x/2xy= 3 3 3 √54a3xy, and so on for the other expressions given in the definition. 15. When surd quantities have different radical signs or indices, they must be reduced to a common index, before their sum, difference, product, or quotient, can be found, in its most simple form, which is only reducing the indices to a common denominator, as in vulgar fractions, and then involving each quantity to the power denoted by its numerator. Thus, if it be required to reduce 3 and 4, to a common index. Here, √3=3, and 4-4; therefore 31⁄2=3*=(33)*—27* ; and 3. 43=48=(42)*—168. Also, if it be required to reduce 4x and 6x to a common index. Here, 4√√x=4x2=4x3=4(x3)3 or 4√x3, and 6x=6x-6x=6(x2) or 64x2. When a surd quantity does not admit of decomposition, it is already in its most simple form. Thus, 15 cannot be reduced lower, neither of its factors, 3 and 5, being squares. Ι Ι Also, reduce x and ym to a common index. 16. The reduction of surd quantities, is of great utility. Thus, if it be required to find the sum and difference of 28 and √63. Here, /28/4×√/7=2√/7, and √63=√9x √7=3√/7; then 2/7 +3√√/7=5√/7 their sum, and 3√/7 −24/7 = 7 their difference: 3 Also, if it be required to find the sum of the following quantities,√16a3 ̧√/200a2, and √/162a2. By reduction 16a32a/2, 200a2 = 10a/2, √/162a29a/2, therefore their sum will be 21a/2. Also, if it be required to find the sum and differ ence of 12x2y and/ 15x2y and These may be reduced to a common denominator, by multiplying the numerator and denominator of the first quantity by 4, and the numerator and denominator of the second quantity 17. If it be required to multiply two surd quantities together, their radical signs or indices being common, it is done by Note to Case III. in Multiplication; but if different, as in Def. 15, they must be reduced to a common index. Thus, if it be required, to multiply 2 by 4; 3 Here, √/2=21—2* = (23)*= 85, and √4=43—4— (42) *—163, (128) Product. 4 Also, if the product of 4/x by be required, Here, 3/4x4, and 4x=4x=4x4; therefore, 3x4×4x4 = 12x4, or 12(x3), or 12. 18. If it be required to divide one surd quantity by another. Thus, divide 9/56 by 3/7. Here, the rational part of the dividend 9, being divided by 3, the rational part of the divisor, gives 3 for the quotient. Thus Also, 56, the surd part of the dividend, being divided by 7, the surd part of the divisor. √56÷√7, √56÷78; therefore, 38 is the quotient required. But if the surds be of different kinds, they must be reduced to a common index, by Def. 15. Thus, divide 3 by 3 9. Here, 3=94–93⁄4, and √9–93—9%; 4 Also, if it be required to divide 123 by 4√y. Here, 12y=12y, and 4 √/y=4y=4y4; 4 3 NOTE. The division of Surds is only subtracting the inde: of the divisor from the index of the dividend. From this i follows, that the denominator of a fraction may be place in the numerator, and vice versa, by changing the sign of it. index. 4. Let x.(x+y) ̄2 be expressed by a positive index. SIMPLE EQUATIONS. EXAMPLES In Simple Equations, with their solutions, in which the preceding definitions and reductions are applied. 1. Given 5x+35=4x+38, to find the value of x. Here (6), by transposition, 5x-4x=38-35; ... by Case II. in Addition, x=3. 2. Given 3x+7-2x=6x-13, to find the value of x. Here (6), by transposition, 7+13=6x+2x-3x; ... by Case II. in Addition, 20=5x; (6) by transposition, 3x-2x-6x=-13-7. then (6. Cor.), 5x=20; .'. (ax. 4) x= 20 5 3. Given 3a +4x=2b-x, to find the value of x. Here (6), by transposition, 4x+x=2b—3a, or 5x-26-3a; (ax: 4.) x= 26-3a N. B. From these solutions it appears, that the unknown quantities may be transposed to either side of the equation, and the known quantities to the other. NOTE.-The Student may prove his questions by substituting the value of a, when found, in the given equation, and if the sides remain equal after substitution, the number substituted will be the value of the unknown quantity; thus, in the first example, x is found to be equal to 3, then will 5a 15, and 4 12; .. if 15 be substituted for 5x, its value, in the given equation, and 12 for 4x, its value, then the equation, which is 5x+35=4x+38, becomes 15+35=12+38, or 50=50; therefore, the value of x, in this case, is 3. |