exponent by the same number. The following operations upon surds depend upon this principle. 85. 1st, To reduce a quantity to the form of a given surd. RULE. Raise the quantity to the power denoted by the exponent of the surd, and indicate the extraction of the same root. EXAMPLE. Reduce 2a to the form of the cube root. Here 2a raised to the third power becomes 8a3, and indicating the extraction of the cube root, (8a3), the form required. 2a 4a2 5a 1. Express each of the quantities, ax, 3ay, '322 and y separately in the form of the square root. 4a2 16a4 25a2 (a2x2)*, (9a2y2)*, (¿=2)3, ('1o¿), (10)3, and / 16c4 За Ans. 16 2. Express each of the quantities, 2a2c, 4ay2, 3a2x, and separately in the form of the cube root. 4ac (8a°c3)3, (27a*)3, (64a3y©)3, (27ɗ31⁄23)†, and 3. Express each of the quantities, —2a, Зап and separately in the form of the fifth y2 243c5x5 164a3c3 8x3 Ans. 3cx ax ay a' y2' — mn2' root. Ans. (-3), (2), (*) (*), and (+) 243αδη 3. 10 уго NOTE. If a surd have a coefficient, the whole may be expressed in the form of a surd, by raising the coefficient to the power denoted by the surd, and multiplying this power into the surd, then placing the symbol of the root over the whole product. Thus, 3√x=9x, 2(acx)=(8acx)3. 4. Express 4√ā, 3/ax2, 5(ax2)‡, and 3(x2y3)‡, in the form of simple surds. Ans. 16a, (27ax2), (625ax2), and (81x2y). 86. 2d, To reduce a surd to its simplest form. RULE. Resolye, if possible, the quantity into two factors, one of which shall be a complete power, the root of which is denoted by the surd; place the root of this factor before the symbol, and it will be the form required. If the surd have a denominator, multiply both numerator and denominator of the fraction by such a quantity as will make the denominator the power denoted by the surd, then extract its root, and place it without the symbol. EXAMPLE. Reduce /27a3 to its simplest form. Here 27a3-9a x3a, the first factor is a square, extracting its root, and placing it without the symbol, we have 3a3a, the form required. 1. Reduce /32a3, 3/81a7, √125, and (180a3x2), to their simplest forms. Ans. 4a2√2a, 3a2 3/3a, 5/5, and 6ax J5a. 2. Reduce 3/1250x1y3, (96a5x2)‡, and (72x3 y3), to their simplest forms. Ans. 5xy/10xy2, 2a(6ax2), and 4. Reduce (3), (5), ('5"), and (b), to 36 ах ay 8 their simplest forms. Ans. (x22), (30a), (21ay)*, and (10xy 22). 87. 3d, To reduce surds having different indices to other equivalent ones having a common index. RULE. Reduce the fractional indices to a common denominator, then involve each quantity to the power denoted by the numerator of its fractional exponent, and over the results place for exponent one for a numerator, and the common denominator for a denominator. EXAMPLE. Reduce (2a)1 and (3a2)} to equivalent surds having a common index. Since, and 3=, the quantities are equivalent to (2a) and (3a2), raising each of these quantities to the power denoted by the numerator of its fractional exponent, they become (8a) and (9a4), which is the form required. 1. Reduce (ac) and 5 to equivalent surds having a common index. Ans. (a2c2) and (125). 2. Reduce 43 and 34 to equivalent surds having a com ac 4. Reduce (~) and (4) to equivalent surds having 3 ас ing a common index. Ans. (5) 4 ax Ans. (***) and (10%). 88. To add or subtract surds. 3a2c RULE. Reduce the surds to their simplest form; then if they have the same radical quantity in each, the sum of the coefficients prefixed to this radical will be their sum; and the difference of the coefficients prefixed to the radical will be their difference. But if they have different radi. cal quantities, their sum can only be indicated by placing the sign plus between them, and their difference by placing the sign minus between them. The reason of this rule is obvious, for the radical quantity may be represented by a letter, and then the rule will be identical with that of addition and subtraction in alge bra. EXAMPLE. What is the sum and difference of √288 and 128. Here √288=√144×2=12√2, and 128 =√64×2=8√2; hence their sum is 20√2, and their difference is 4√2. 1. Find the sum and difference of 3√32 and 2162. Ans. sum 30/2, diff. 6√2. 2. Find the sum and difference of 3,3/54 and 3/250. Ans. sum 143/2, diff. 43/2. 3. Find the sum and difference of 3/24a4 and/192a. Ans. sum (2a+4)/3a, diff. (2a-4)/3a. 4. Find the sum and difference of 80 and √45. Ans. sum 75, diff. √5. 1 5. Find the sum and difference of (363) and (98a)*. Ans. sum (6a√a+7√2a), diff. (6a√a—7√2a.) 6. Find the sum and difference of (1000a3) and . (300a3). Ans. sum (10a210a+10a√3a), and diff. 10a2/10a -10a√3a. TO MULTIPLY SURDS. 89. RULE. Reduce the surds, if necessary, to a common index, then multiply the coefficients together for a coefficient, and the surd quantities together for the surd, over which place the common index. EXAMPLE. Multiply 3/10 by 2 3/12. 3x2=6, the coefficient, and 3/10/12/120. .. 63/120 is the result; which, however, can be simplified; for 120-3/8x15=23/15; hence the quantity in its simplest form is 123/15. EXAMPLE 2. Multiply ✔a by 3/b. а Here √a=a1=a¤=(a3)3, and‚3/õ=(b)3=b&—(b2)& Ans. 30/10. Ans. 103/9. Ans. (233255) or 225000. 90. RULE. Reduce the quantities, if necessary, to a common index, then divide the coefficients and the surd quantities separately as in rational quantities. EXAMPLE. Divide abac by b3/bc. Here ab÷ba, the coefficient, and ac÷be the surd 3/a .. the quotient is a EXAMPLE 2. Divide 31 ac by 23/bc. Here the quantities reduced to a common index become 3(43ç3), and 2 (b2c2)ể ..the coefficient of the quotient is, and the surd which reduced to its simplest form is (ab1c), and hence the quotient is 26 91. RULE. Raise the coefficient of the surd to the required power, and then multiply the exponent of the surd by the exponent of the power. EXAMPLE. Find the third power of 21/ac. Here we raise 2 to the third power, which gives 8, and then multiply, the exponent of the surd, by 3, the expoBent of the power, which gives .. the third power of 3 2 2√āc is 8(ac)312=8 (a2c2 × ac)=8ac(ac) or 8ac√āc. ac 1. Raise 2(ac) to the second power. 2. Raise 4(bcx2) to the third power. 3. Raise 6 to the fourth power. Ans. 4(ac)3. Ans. 64bc3 bc. Ans. 1+3√x+3x+x√x. 6. Raise (3+2√5) to the second power. Ans. 29+12√5. |