If there is now no remainder, the root has been found. If there be a remainder, consider the two terms of the root already found as one and proceed as before. 224. The following examples worked out will make the process more clear. Here the second term of the root, and consequently the second term of the divisor, will have a negative sign prefixed, Next take a case in which the root contains three terms. a2+2ab+b2—2ac−2bc+c2 (a+b−c When we obtained the second remainder, we took the double of a+b, considered as a single term, and set down the result as the first part of the second divisor. We then divided the first term of the remainder, -2ac, by the first term of the new divisor, 2a, and set down the result, c, attached to the part of the root already found and also to the new divisor, and then multiplied the completed divisor by -C. Similarly we may proceed when the root contains 4, 5 or more terms. EXAMPLES.-LXXX. Extract the Square Root of the following expressions: 1. 4a2+12ab+9b2. 6. x2-6x3+19x2 - 30x+25. 2. 16h0–247573+97. 7. 9x+12x3 +10x2+4x+1. 3. a2b2+162ab+6561. 8. 44-12p3+13r2 −6r+1. 4. y6-38y3+361. 9. 4n+4n3-7n2 - 4n+4. 5. 9a2b2c2-102abc+289. 10. 1-6x+13x2 — 12x3 + 4.x1. 11. x-4x+10x1—12x3+9x2. 12. 4y-12y3z+25y222-24yz3+162. 13. a2+4ab+4b2+9c2 + 6ac +12bc. 14. a6+2ab+3a4b2+4a3b3 +3a2b1+2ab5+ bo. 15. x-4x+6x3+8x2+4x + 1. 16. 4x4+8ax3 +4a2x2+16b2x2+16ab2x+16ba. 17. 9-24x+58x2-116x3 +129x1-140x+100xo. 18. 16a-40a3b+25a2b2 — 8ab2x+64b2x2+64a2bx. 19. 9a1-24a3μ3 —30a3t +16a2p¤ +40ap3t+25t3. 225. When any fractional terms are in the expression of which we have to find the Square Root we may proceed as in the Examples just given, taking care to treat the fractional terms in accordance with the rules relating to fractions. and then take the square root of each of the terms of the fraction, with the following result: b4 + 226. XVIII. ON CUBE ROOT. THE CUBE ROOT of any expression is that expression whose cube or third power gives the proposed expression. Thus a is the cube root of a3, 36 is the cube root of 273. The cube root of a negative expression will be negative, for since (− a)3 — — a × − a × − a = − a3, the cube root of a3 is · -α. and So also The symbol the cube root. -3x is the cube root of -27.x3, -4a2b is the cube root of -64ab3. is used to denote the operation of extracting 227. We now proceed to investigate a Rule for finding the Cube Root of a Compound Algebraical Expression. We know that the cube of a + b is a3+3a2b+3ab2+b3, and therefore a+b is the cube root of a3+3a2b+3ab2+b3. We observe that the first term of the root is the cube root of the first term of the cube. Hence our rule begins: "Arrange the terms in the order of magnitude of the indices of one of the quantities involved, then take the cube root of the first term and set down the result as the first term of the root: subtract its cube from the given expression, and bring down the remainder:" thus a3+3a2b+3ab2+b3 (a a3 3a2b+3ab2+b3 Now this remainder may be represented thus, b (3a2+3ab+b2) ; hence if we divide 3a2b+3ab2+b3 by 3a2+3ab+b2 we shail obtain +b,-the second term of the root. Hence our rule proceeds: "Multiply the square of the first term of the root by 3, and set down the result as the first term of a divisor :" thus our process up to this point will stand thus: |