a*—2a3x+3a3x2—2ax3+x1(u2—ax+x3, a1 2a3)—2u3 x aa—2a3x+a3x' 2a2)2a2x2 a^—2a3 x2-3a2x2 —2a. 3+xa. 2. Extract the cube root of x+6x-40x3 +96x-64, x® +6x3 —40x3 +96x—64(x2+2x-4 x6 3x4)6x5 x+6x+12x2+8x3 3x1)—12x1 x+6x3-40x3 +96x-64 3. Required the square root of a2+2ab+2ac+6°+2bc+c*. Ans. a+b+c. Ans. -2x+1. 4. Required the cube root of x-6x+15x*—20x3 +-15 x2-6x+1. 5. Required the biquadrate root of 16u*-95u3x+÷16 a2x2-216ax 3 +81x1. Ans. 2a-3x, SURDS. SURDS are quantities, which are not exact roots, being usually expressed by the powers with fractional indices, or by means of the radical sign ✔. Thus, 24, or √2, which denotes the square root of 2. of And 3 or 3, signifies the cube root of the square 3; where the numerator shows the power, to which the quantity is to be raised, and the denominator its root. 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 the form required. EXAMPLES. 1. To reduce 3 to the form of the square root. First 3×3-23-9; then ✔ 9 is the answer. 2. To reduce 2x to the form of the cube root. First, 2x2 ×2x* <2x2=2x3 3 3 =8x6 Then ✓ 8x, or 8x13, is the answer. 3. Reduce 5 to the form of the cube root. Ans. 125, or 125. 4. Reduce xy to the form of the square root. 5. Reduce 2 to the form of the 5th root. Ans. Vix3y. Ans. 32. 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 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 numera tor. EXAMPLES. 1 1. Reduce 15 and 9 to equivalent quantities, having the common index. +=+=+=+=+= the first index. Therefore 15 and 9*are the quantities required. PROBLEM III. To reduce surds to their most simple terms. RULE.* Resolve the given surd into two factors, one of which shall be the greatest power of the corresponding denomination, contained in the surd, that can be one factor, and 'set its root before the other factor, with the proper radical sign between them. EXAMPLES. 1. To reduce 48 to its most simple terms. 48=√16x3=√16x3=4xV3-4V3 the answer. 2. Required to reduce $ 108 to its most simple terms. √108=3√27x4=3√27×3√4=3x3√4=33√4 the an 1. Reduce such quantities, as have unlike indices, to other equivalent ones, having a common index. When the given surd contains no exact power, it is already in its most simple terms. Vol. I. M m 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 quantities, they can only be added by the signs + and -. EXAMPLES. 1. It is required to add ✔ 27 and 48 together. First, 27=V9 3=3√3; Then, 3√3+43=3+4x√3=7√/3= sum required. 2. It is required to add 3500 and $108 together. First, 500-3125x453√4; And 3/108-3v 27x4=34; Then, 53/4+33√4=5+3×3√4=83/4= sum required. 3. Required the sum of 72 and √128. 4. Required the sum of 27 and 147. 5. Required the sum of and. Ans. 14V/2. Ans. 103. 6. Required the sum of 340 and $135. 7. Required the sum of 3 and 3 PROBLEM V. 2 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 |