5. Required the square root of a2---x* 1. Find the root of the first term, and place it in the quotient. 2. Subtract the power, and bring down the second term for a dividend. 3. Involve the root, last found, to the next inferior power, and multiply it by the index of the given power for a divisor. 4. Divide the dividend by the divisor, and the quotient will be the next term of the root. 5. Involve the whole root, and subtract and divide as before; and so on, till the whole be finished. EXAMPLES. 1. Required the square root of a1—2a3x+3a a2 — ·2a3x+3a2x2——2ax3+xa (aˆ—axfx2· a4 2a2)-2a3 x 2 a 2a2)2a2x2 -2a3x+3a2x2 2. Extract 2. Extract the cube root of x+6x5—40x3 +96x-64, x+6x3—40x3 +96x——64(x2+2x−4 મ 3. Required the square root of a+2ab+2ac+b2+2b0 +c2. Ans. a+b+c 4. Required the cube root of x-6x+15xa—20x3+ 1jx2-6x+1. Ans. x-2x+!2 5. Required the biquadrate root of 16a *—96a3x-216 x2-216ax3--81x+. Ans. 2034 SURDS, SURDS are such quantities as have no exact root, being usually expressed by fractional indices, or by means of the radical sign✔✅ Thus, 2, or 2, which denotes the square root of 2. 3 And 33, or 3*, signifies the cube root of the square of 3; where the numerator shews the power, to which the quantity is to be raised, and the denominator its root. PROBLEM PROBLEM I. To reduce a rational quantity to the form of a surd. RULE Raise the quantity to a power equivalent to that, denoted by the index of the surd; then over this new quantity place the radical sign, and it will be of the form required. EXAMPLES, 1. To reduce 3 to the form of the square root. First 3X339; then 9 is the answer, 2. To reduce 2x2 to the form of the cube root. 3. Reduce 5 to the form of the cube root. I Ans. 12513, or 125. 4. Reduce xy to the form of the square root. 3/125. Ans.***. ོ་ Reduce 2 to the form of the 5th root. Ans. 3213. PROBLEM II. To reduce quantities of different indices to other equivalent ones, that shall have a common index. RULE. 1. Divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities. 2. Over 2. Over the said quantities with their new indices place the given index, and they will make the equivalent quantities required. NOTE. A common index may also be found by reducing the indices of the quantities to a common denominator, and involving each of them to the power, denoted by its numerator. EXAMPLES. 1. Reduce 15 and 9 to equivalent quantities, having the common index. I 2 4÷÷÷÷=1X=1= the first index. ==== the second index. Therefore 153|* and 91* are the quantities required, 2. Reduce a and x to the same common index + 4. Reduce a and b to the common index . PROBLEM III. To reduce surds to their most simple terms. RULE. Find the greatest power contained in the given surd, and set its root before the remaining quantities, with the proper radical sign between them. EXAMPLES. i. To reduce 48 to its most simple terms. √48/16X3=√/16X✔✅✔✅/3=4×34/3 the answer. 3 3 2. Required to reduce 108 to its most simple terms. 3 10827X427X4=3X434 the Ans. 7a 2x. 7. Reduce 984"x to its most simple terms. PROBLEM *When the given surd contains no exact power, it is already in its most simple terms. |