:1 :: · 341883: 0051679. And consequently x=10·6-·0051679=10.5948321, very nearly. EXAMPLES FOR PRACTICE. 1. Given 3+10x2+5x=2600, to find a near approximate value of x. 2. Given 2x4-16x3+40x2-30x+1=0, to find a near value of x. 3. Given value of x. Ans. x 11.00675. Ans. x 1.284724. Ans. 8 414455. Ans. 4510661. 2+2x1+3x3+4x2+5x=54321, to find the (73+4x2)+√(20x2-10x)=28, to find the 5. Given √{144x2− (x2+20)o}+√{196.x2 - (x2+24)2}= 114, to find the value of x. Ans. 7.123883. OF EXPONENTIAL EQUATIONS. An exponential quantity is that which is to be raised to some unknown power, or which has a variable quantity for its index, as And an exponential equation is that which is formed between any expression of this kind and some other quantity, whose value is known; as a=b, xa, &c. where it is to be observed, that the first of these equations, when converted into logarithms, is the same as x log. a= log. b, or x= And the second aa, is the same as x log. x = log. a. log. b log. a' In the latter of which cases, a near approximate value of the unknown quantity may be determined, as follows • RULE. Find, by trial, two numbers as near as can conveniently be done to the number sought, and substitute them in the given equation, x log. x = log. a, instead of the unknown quantity, noting the results obtained from each, as in the rule of Double Position, before laid down. Then, by means of a certain number of successive operations, performed in the same manner as is there described, the value of x may be found to any degree of accuracy required*. 1. Given EXAMPLES. 100, to find an approximate value of x. Here, by the above formula, we have x log. x = log. 100 = 2. And since x is readily found, by a few trials, to be nearly in the middle between 3 and 4, but rather nearer the latter than the former, let 3.5 and 3'6 be taken for the two assumed numbers. Then log. 355440680; which, being multiplied by 3.5, gives 1.904238 first result. And log. 3 65563025; which, being multiplied by 3.6, gives 2.002689 for the second result. for the first correction; which, taken from 3.6, leaves x=3 5927, nearly. * Many attempts have been made to determine the value of the unknown quantity, in the exponential equation a=a, above given, by converting it into a series, the terms of which shall consist only of a and its powers; but no expression of this kind has hitherto been discovered, which is sufficiently convergent to answer any practical purpose. See Vol. II. of my Treatise on Algebra, before referred to. And as this value is found, by trial, to be rather too small, let 3.59727 and 3.59729 be taken as the two assumed numbers. Then log. 3 597275559731; which, being multiplied by 3 59727, gives 1.9999854 = first result. And log. 3 59728=5559743; which, being multiplied by 3.51728, gives 1·9999952 = second result. Whence 1.9999952..3 59728..2. 1.9999854..3 59727..1.9999952 ·0000098 00001:: 0000048: 00000485 for the second correction; which, added to 3.59728, gives x=3 59728485, the answer required; being a value of x extremely near the truth. 2. Given 2000, to find an approximate value of x. Ans. x=4 82782263. 3. Given (6x)=96, to find an approximate value of x. Ans. x 1.8826432. 4. Given 123456789, to find an approximate value of x. Ans. 8 6400268. 5. Given x* — x—(2x — x′′), to find an approximate value of x. Ans. x 1.747933. OF THE BINOMIAL THEOREM. The binomial theorem is a general algebraical expression or formula, by which any power, or root of a given quantity, consisting of two terms, is expanded into a series, the form of which, as it was first proposed by Sir I. Newton, being as DQ + EQ, &c.] Where p is the first term of the binomial, Q the second term m - divided by the first, the index of the power, or root, n and A, B, C, &c., the terms immediately preceding those in which they are first found, including their signs + or -. This theorem may be readily applied to any particular case, by substituting the numbers, or letters, in the given example, for P, Q, m, and n, in either of the above formulæ, and then finding the result according to the rule.* This celebrated theorem, which is of the most extensive use in algebra, and various other branches of analysis, may be otherwise expressed as follows: It may here also be observed, that if m be made to represent any whole or fractional number, whether positive or negative, the first of these expressions may be exhibited in the more simple form m(m-1)(m-2)..... [m-(n-1)]a"x"=" Where the last term is called the general term of the series, because if 1, 2, 3, 4, &c., be substituted successively for n, it will give all the rest. |