33. °1/64, °1′ ́6.4, °1 0.64, 1/0.064, 10.0064, °1 0.00064. 6 34. v 74300, V7430, V743, '√74.3, °√/7.43, °√/0.743. 35. Having found four figures of the cube roots of the numbers in examples 3, 4, 6, 11, and 13, find three more figures in each of these roots by 301. 36. 282429536481. 37:208827064576. 302. The nth Root of a Polynomial.-A rule for finding the nth root of a polynomial can be obtained by observing the formation of the nth power of a polynomial, a being any integral number whatever (2265). th root of One observes that the first term of the root, a, is the n' a", the first term of the quantity whose root is to be found, and that the second term of the root, b, is found by dividing the second term of the given quantity, nan-1b, by nan-1, or by n times the (n-1)th power of the first term of the root. If the root now found be raised to the nth power, and subtracted from the quantity whose root is desired, it will be apparent that the two terms of the required root have been found. NOTE. The student should observe that the process just described is an extension of that discussed for the square and cube roots of polynomials and numbers. RULE. Arrange the terms according to the powers of some letter. Find the required root of the first term of the given polynomial, for the first term of the root, and subtract the root now found raised to a power equal to the required root, from the given polynomial. Divide the first term of the remainder by n times the (n of this root for the second term of the root and subtract the n' the root now found from the given polynomial. th 1)th power power of If there is a remainder, use the same trial divisor as before, and proceed in like manner till the nth power of the root becomes equal to the given polynomial. EXERCISE LV 1. Find the fourth root of 8-4 x 10 x 16 x 19 x* 1610x-4x+1. x-4x2+10x6 — 16 x2+19 x1 — 16 x3+10x2-4x+1 | x2 −x+1 (x2−x+1)1 = = x8 · 4 x2+10x6—16 x2+19 x1−16 x3+10x2-4x+1 2. Extract the fifth root of 32 x5 — 240 x1 + 720 x3 3. Extract the fourth root of Ans. x2-x+1. x-4x+18x-40x5+91 x110x3-54x2+ 108x+81. -- 4. Find the development of x1 − x3 + x2 − x + 1 to 4 terms. 5. Extract the fifth root of x2 + x1 + x3 + x2 + x + 1 to 3 terms. 6. Show that the fifth root of 243.1 is 3.00024. 7. Find the fifth root of 1024.68 and 16805.81 correct to 5 decimals. CHAPTER III FRACTIONAL AND NEGATIVE EXPONENTS 303. am, when m is a positive integer (? 9, 3) has been defined by the equation In other words, am is an abbreviated way for writing the product of m factors, each equal to a. 304. It has been shown that positive integral exponents obey the five laws expressed by the following equations: Thus far an exponent has been regarded as a positive integer; but it is very advantageous to use exponents which are not positive integers. The meaning of such exponents will now be shown. 305. No definition has been given of fractional and negative indices, so any definitions may be given to them; since the law for positive integral exponents has already been obtained, expressed by the equation, (I) aman am+n it would be most natural and most convenient to give fractional indices and negative indices such definitions as will make the important relations I-V always true whatever m and n may be. is required. Since relation I is Thus at is such a number that in For example, the meaning of a to be fulfilled, a§ · a§=a!+ != a. case it is multiplied by itself the product is a; but the square root of a is such a number. Therefore a must be equivalent to the square root of a; that is, al = Va. Similarly, what is the meaning of at? If relation I is fulfilled, then These examples will show the student what is meant by frac In the next four articles the definition of the law which fractional and negative exponents obey will be given in general symbols. 1 1 306. Required the meaning of a" when n is any positive integer. By hypothesis, + 1 + to n terms = an = a; hence a" must be equivalent to the nth root of a; that is, m 307. Obtain the meaning of a" where m and n are any positive Hence, when m and n are positive integers, a" means the nth root of the mth power of a; that is, in the case of a positive fractional exponent, the numerator denotes a power and the denominator a root. 308. A meaning has now been given to the terms, positive integral exponent and positive fractional exponent; and it remains to assign a meaning to negative exponents whether they be integral or fractional. For example, it is required to find the meaning of a2. If the law expressed by 304, I, always holds, whatever m and n may be, then by hypothesis, The definition may now be formulated in general symbols. 309. What is the meaning of a-" when n is any positive number, whole or fractional? By hypothesis, whatever m may be, Suppose that m is positive and that m>n; then by 304, I, Hence, a" is the reciprocal of a"; or this result may be written symbolically in any of the following ways: 310. It will follow from the meaning given to a negative index that am÷a" = am¬n when m is less than n as well as when m is greater than n. let m be less than n, then For, 311. Thus, for example, according to the preceding definitions, |