8. Required to find the mth power of a PROBLEM IX. To Evolve or Extract the Roots of Surd Quantities*. Extract both the rational part and the surd part. Or divide the index of ihe given quantity by the index of the root to be extracted, then to the result annex the root of the rational part, which will give the root required. EXAMPLES. 1. Required to find the square root of 16 V 6. Firsi, v 16 =.4, and €64, 3 theref. 116 v 6, = 4.61 = 496, is the sq. root required. 2. Required to find the cube root of z'z v 3. First, = }, and (V 331 31=3 = 36; theref. x'q 3)) = = 4.36 = } %3, is the cube root required. 3. Required the square root of 63. Ans. 6 ✓ 6, 4. R:quired the cube root of fa3b. Ans. La for 3. Required the 4th root of 16a?. Ans. 2 van 6. Required to find the mth root of x 7. Required the square root of a? - 6a V b + 9b. 2 2 The square root of a binomial or residual surd, a + b, or & -b may be found thus : Take a -62 = 0; a toc then vatb=v-tv etc and ✓a-b= ✓ 2 2 Thus the square root of 4 + 23=1tv3; and the square rout of 6 2 v 5 ✓ 5 - 1. But for the cube, or any higher root, no general rule is known. INFINITE INFINITE SERIES. An Infinite Series is formed either from division, dividing by a compound divisor, or by extracting the root of a compound surd quantity; and is such as, being continued, would run on infinitely, in the manner of a continued decimal fraction. But, by obtaining a few of the first terms, the law of the progression will be manifest ; so that the series may thence be continued, without actually performing the whole operation. PROBLEM I. To Reduce Practional Quantities into Infinite Series by Division. Divide the numerator by the denominator, as in com non division; then the operation, continued as far as may be thought necessary, will give the infinite series required. 463 &c. a 8. Expand into an infinite series. (a+b)* 26 363 Ans. I + a2 23 1 7. Expand -= 1, into an infinite series. 1+1 PROBLEM II. To Reduce a Compound Surd into an Infinite Series. EXTRACT the root as in common arithmetic; then the operation, continued as far as may be thought necessary, will give the series required. But this method is chiefly of use in extracting the square root, the operation being too tedious for the higher powers. EXAMPLES EXAMPLES. 1. Extract the root of as ora in an infinite series. 2. Expand v1+1=v2, into an infinite series. Ans 1 + $-+- tis &c. 3. Expand ✓ 1 i into an infinite series. Ans. I-}-}-is - Tis &c. 4. Expand va? + rinto an infinite series, 5. Expand vao - 20 * to an infinite series. PROBLEM III. To Extract any Root of a Binomial : or to Reduce a Binomial Surd into an Infinite Series. This will be done by substituting the particular letters of the binomial, with their proper signs, in the following general theorem or formula, viz. m - 2n (P + PQ) "=p"+-AQ+ BQ co kc. 2n and m n 3n and it will give the root required : observing that p denotes the first term, 4 the second term divided by the first, the index of the power or root ; and A, B, C, D, &c, denote the several foregoing terms with their proper signs. EXAMPLES. m n m--n 2a 1. To extract the sq root of q? + 63, in an infinite series. 63 Here P = a’,4 =- and s-i therefore a? 2 62 62 a 2a 64 BQ X -X-X = C, the 3d term. 2n 2.423 366 =o the 4th. Зп % 423 2.4.625 62 6 3.66 Hence at + 66 568 &c. is the series required. 2a 8a3 1645 128a7 1 2. To find the value of or its equal (a-x)-, in an infinite series. (a-x) 64 &c. or Note. To facilitate the application of the rule to fractional ex. amples, it is proper to observe, that any surd muy be taken from the denominator of a fraction and placed in the numerator, and vice versa, by only changing the sign of its index. Thus, 1 1 1X*-2 or only x 2; and =1x (a + b)-2 or (a + b)2 3 (a + b)-1; and =a? (a + x)*2; and ) Here X 1 *xx}; also |