Find to four decimal places the cube root of: 241. Since the fourth power is the square of the square, and the sixth power the square of the cube, the fourth root is the square root of the square root, and the sixth root is the cube root of the square root. In like manner, the eighth, ninth, twelfth, ..... root may be found. EXERCISE 89. Find the fourth root of: 1. 81 x2 + 108 x3 + 54 x2 + 12 x + 1. 2. 16 x2 - 32 ax3 + 24 a2x2 - 8 a3x + a*. 3.1+4x + x + 4x2+10x + 16x8 + 10x2 + 19x2+16x5. Find the sixth root of: 4.1+6d+ d° + 6 d5 + 15 d2 + 20 d3 + 15 d2. 5.729-1458 x + 1215 x2 — 540 x2 + 135 x4 – 18 x5 + xo. 6.1-18y + 135 y2 - 540 y3 + 1215 y2 – 1458 y5 + 729y. Find the eighth root of: 7.1-8y+28 y2 56 y3 +70y - 56 y5 + 28 yo – 8 y2 + y2. CHAPTER XVI. THEORY OF EXPONENTS. n 242. If n is a positive integer, we have defined an to mean the product obtained by taking a as a factor n times, and called an the nth power of a; we have also defined Va as a number which taken n times as a factor gives the product a, and called Va the nth root of a. n 243. By this definition of a" the exponent n denotes simply repetitions of a as a factor; and such expressions as a, as have no meaning. It is found convenient, however, to extend the meaning of a" to include fractional and negative values of n. 244. If we do not define the meaning of a" when nis fractional or negative, but require that the meaning of an must in all cases be such that the fundamental index law shall always hold true, namely, am X an = am+n, we shall find that this condition alone will be sufficient to define the meaning of a" for all cases. 245. Meaning of a Zero Exponent. By the index law, Therefore, the zero power of any number is equal to unity. 246. Meaning of a Fractional Exponent. By the index law, ton terms an X an... mn an = am; m m ..... to n factors = a" provided m and n are positive integers. The meaning, therefore, of a", where m and n are positive integers, is the nth root of the mth power of a. Hence, The numerator of a fractional exponent indicates a power and the denominator a root. 247. Meaning of a Negative Exponent. By the index law, if n is a positive integer, But an xa-n = an-n = ao. (§ 245) ... an Xa-" = 1. That is, a" and an are reciprocals of each other (§ 169), 248. Hence, we can change any factor from the numerator of a fraction to the denominator, or from the denominator to the numerator, provided we change the sign of its exponent. ab2 c3d3 Thus, may be written ab2c-3d-3, or 1 249. We have now assigned definite meanings to fractional exponents and negative exponents, by assuming that the index law for multiplication, am × a" = am + ", is true for all values of the exponents m and n. It remains to show that the index laws established for division, involution, and evolution apply to fractional and negative exponents. 250. Index Law of Division for all Values of m and n. To divide by a number is to multiply the dividend by the reciprocal of the divisor. Therefore, for all values of m and n, 251. Index Law of Involution and Evolution for all Values of m and n. To prove (am)n = amn for all values of m and n. CASE 1. Let m have any value, and let n be a positive CASE 3. Let m have any value, and n = - r, r being a positive integer or a positive fraction. Therefore, (am)n = amn for all values of m and n. 252. To prove (ab)" = anb" for any value of n. CASE 1. Let n be a positive integer. Then, (ab)" = ab × ab × ab ..... to n factors (§ 247) = (a xa..... ton factors) (b× b ..... to n factors) = anbn. CASE 2. Let n = 2, , p and q being positive integers. Then, by Case 1, § 251, since q is a positive integer, the qth p p power of (ab) = (ab) x (ab) ..... to q factors Also by Case 1, § 251, since q is a positive integer, the qth But the 9th power of (ab) = (ab)2 = a2br. Extracting the qth root of each member, we have CASE 3. Let n = -r, r being a positive integer or fraction. Then, 1 1 arbr = a-rb-r. (ab)-* = (ab) |