be extracted, and the numerator the power to which the quantity is to be raised. Such quantities may be represented without fractional indices, as may be found convenient; that is, x by x3, a* by 3⁄4a, a" by "JaTM, &c. m It remains therefore to show, that ao a1a? + n РЯ =α multiplied mq times into itself 1 =a" multiplied np times into itself 1 .. ao a1 =ɑo1 multiplied (mq+np) times into itself 5 Examples. √xxx=x+++=x& √x÷3x=x++=x+ 110. Any power or root of a quantity with a fractional index will be expressed, by multiply. ing or dividing the index of the quantity by the index of the power to which it is to be raised, or root to be extracted. 111. Defs.-Surds or irrational quantities are such as do not admit of an exact root, as √2, √3, (a+b), &c. Other quantities are called rational ones. A multiplier which is not within the radical sign, or parenthesis (), is called the coefficient of the surd. Thus 5 is the coefficient of 5/2, &c. Surds are said to be similar, that I have the same quantity under the radical sign ; thus 4/3, 8/3 are similar surds. 112. A rational quantity may be represented in the form of a surd, by raising the quantity to the power whose root the surd expresses, and affixing the radical sign. Thus x may be repre sented 3 and 5 by √25, 1 n х =xmn, &c. 113. The root of a product is equal to the product of the corresponding roots of the factors of 1 = that product, that is Ja× √b=√ab, and a"b" — 1 1 (ab)"; since each of the quantities ab and (ab)" when raised to the nth power, is equal to ab. 114. Surds can be reduced to more simple forms when the parts under the radical sign can be divided into 2 factors, one of which is a power corresponding to the root of the surd. Thus √x2y = √x2 × √y (Art. 113) =xy; again, √ a2 — a2 x = √ a2 × √1−x=α√/1−x. a EXAMPLES. Reduce ac23 to the most simple form. √α*c*x*=√a*c*x*× √x=acx↓x Reduce 48 to the most simple form. ADDITION AND SUBTRACTION OF SURDS. 135 Reduce 108 to the most simple form. 3/108 3/27 x 3/4 33/4 Reduce 3/320 to the most simple form. 3/320=3/64 × 3/5=43/5 ADDITION AND SUBTRACTION 115. Reduce the surds to their most simple forms. If the surds are similar, add or subtract their coefficients, and affix the common surd to the result. If the surds are not similar, and cannot be reduced to such, connect them with their respective signs + or −, since their sum or difference cannot be more simply expressed. 1. 2. EXAMPLES. Find the sum of √48 and √75. √48=√16 × √3=4√3 √75=√25 × √3=5√3 ..√48+√75=9/3 From 3/448 take 3/189. 3/448 3/64 x 3/7=43/7 3/189/27x3/7=33/7 3/448-3/189-3/7 3. 4. From 7.3/2 take 23/54. 23/54 23/27 × 3/2=63/2 .. 73/2-23/54-3/2 Find the sum of Ja3y and Jab3y. √ab3y=√b2× √ay=bJay :. √a3y+√ab3y=(a+b)√ay MULTIPLICATION AND DIVISION 116. Bring the indices of the surds to a common denominator (if they have not one already), raise each quantity to the power denoted by the numerator of its index, and express the root of each by means of the common denominator; the surds are thus reduced to equal ones having the |