Ex. 6. ao—6a3x+15a1x2—20a3x3-15a2x2—6ax3+x^(a3—3a2x+3ax2-x2 αδ 2a3-3a2x)—6a3x+15a1x2 -6ax 9a1x2 2a-6ax+3ax) 6ax-20a3x+15ax 6a2x2-18a3x3 9a2x1 2a3—6a2x+6ax2—x3)— 2a3x3+ 6a2x2-6x+x6 -2a6a2x-6x+x 7. Extract the square root of 1±x. 1 16. Extract the square root of 4x2-16x3+24x2-16x+4. Ans. 2x4x+2. 17. Extract the square root of 16x*+24x+89x2+60x+100. 18. Extract the square root of 1±y. Ans. 42+3x+10. Ans.1± y y y3 5y+ 8 ±16 128+, &c. 19. Extract the square root of 9x-12x2+10x-28x3 +172 -8x+16. Ans. 3x-2x+x-4. To extract the Cube Root of a Compound Quantity. a3 3a2+3ab+b2)3a2b+3ab2+b3 3a2b+3ab2+b3 56. RULE. Arrange the terms according to the dimensions of some letter, as in division, and extract the root of the first term, which must be a cube. Place this root in the quotient, subtract its cube from the first term, and there a3+3a2b+3ab2+b3(a+b will be no remainder. Bring down the three next terms for a dividend, and put three times the square just found in the divisor's place, and see how often it is contained in the first term of the dividend, and the quotient is the next term of the root. Add three times the product of the two terms of the root, plus the square of the last term, to the terms already in the divisor's place, and the divisor will be complete. Multiply the divisor by the last term of the root, subtract the product from the dividend, and bring down the next three terms for a dividend, and proceed as before. {(axa)3+3(ab)+(bxb)}=3a2+3ab+b2 the first divisor. {(a+b)3×3+3(a+b)c+cXc}=3a2+6ab+362+3ca+3cb+c2, the divisor for the third letter in the root. 3(a+b+c)2+3d(a+b+c)+(dxd)=divisor for the fourth term of the root, &c. &c. 1. Extract the cube root of x2-6x+15x1—20x3+15x2-6x+1. x-6x+15x-20x+15x2-6x+1(x2-2x+1. ၄၆ 3x-6x+4x-6x+15x-20x3 -6x3+12x1— 8x3 3x-12x+15x-6x+1|3x-12x+15x2-6x+1 3x-12x15x2-6x+1 2. Extract the cube root of 2+6x-40x96x-64. x+6x3—40x3+96x—64(x2+2x-4 x6 3x*+6x3+4x2\6x3—40x3 |6x3+12x2+8x3 3x+12x-24x+16—12x-4823+96x-64 -12x1—48x3+96x-64 3. Extract the cube root of 27-54x+63x-71x+57x5-36x+22x3-9x & Root or quotient 32-2x+x+1 27x-54x+63x-71x+57x-36x+22x-9x+3x-1 4. Extract the cube root of a+3a2b+3ab2+b+3a2c+6abc+3b2c+3ac2 + Ans. a+b+c. EVOLUTION. If the root consists of three terms, a, b, c, they may be obtained by first finding a and b, as above, and then deriving c from (a+b) in the same manner that b was derived from a. (a+b)3+3(a+b)3c+3(a+b)c2+c3(a+b+c (a+b)3 3(a+b)2+3(a+b)c+c2)3(a+b)2c+3(a+b)c2+c3 ́3(a+b)2c+3(a+b)c2+c2 6. Extract the cube root of 8x+-362+54x+27. Ans. 2x+3. 7. Extract the cube root of 27x6-5425+63x2-44x3+21x2-6x Ans. 3x2-2x+1. +1. [From p. 41.] Here the numerator being the least compounded, and b rising therein to a single dimension only, I divide the same into the parts 6a3 4a3, and 15ab 10a2bc, which, by inspection, appear to be equal 2a (3a-2c2), and 5a2b(3a2 — 2c2); .. 3a2-2c is a divisor to both parts, and likewise to the whole, expressed by (3a2-2c2)×(2a3 +5a2b); so that one of these two factors, if the fraction given can be reduced to lower terms, must also measure the denominator; but the former will be found to succeed: thus 3a2-2c2)9a3b-27a2bc-6abc2+18bc3 (3ab-9bc .. 3a2—2c2. Here, the denominator being the least compounded, and d rising therein to a single dimension only, I divide the same into the parts 4ad-4acd, and -2ac2+2c3: which, by inspection, appear to be equal to 4ad(a—c), and 2c2 (a—c). .. a-c is a divisor to both the parts, and likewise to the whole, expressed by (4a'd 2c) X (ac); so that one of these two factors, if the fraction given can be reduced to lower terms, must also measure the numerator; but the latter will be found to succeed: thus, a-c)a'd2-c2d2-a2c2+c(ad2+cd2-ac2-ca; .. a-c is the greatest a2d2-c2ď2-a2c2 to 4a2d-4acd-2ac2 203 = ad2+cd-ac2-c3 the Answer. Solution to the 8th. Multiplying the numeator by 5 and I have. 15x-2x+10x-x+2)45x+10x+20x-5x +40(3x 45x5 6x*+30x3 3x2+6x × by 5, 6x1—20x3+23x2—11x+5 15x*—2x2+10x2-x+2)30x*-100x+115x2-55x+25(+2 30x-4x+20x2-2x +4 -96x3-95x2—53x+21 Multiply the last divisor by 32, and we shall have -96x3+95x2-53x+21(480x*—64x3+320x2-32x+64(—5x `480x1—475x3+265x2—105x mult by 32, 411x3 +55x2+73x+64 —96x3+75x2—53x+21)13152x3+1760x2+2336x+2048(—137 1315213015x+7261x-2877 Divide this by 4925, it becomes 323 er operation exactly divides-9623+95x2-51x +21; and.. in the greatest common measure and the reduced form, is SIMPLE EQUATIONS. 3x3+x2+1 5+x+2 1. An equation is a proposition which declares the equality of 2 quantities expressed algebraically. This is done by connecting these quantities by the sign (=): thus, x-4-6-x is an equa |