218. From the Law of Signs in multiplication it is evident that all even powers of a number are positive; all odd powers of a number have the same sign as the number itself. Hence, no even power of any number can be negative; and the even powers of two compound expressions that have the same terms with opposite signs are identical. 219. Binomials. By actual multiplication we obtain, (a + b)2 = a2 + 2 ab + b2; (a + b)3 = a3 +3 a2b +3 ab2 + b3; (a+b)* = a* + 4 a3b+6a2b2 + 4 ab3 +ba. In these results it will be observed that: 1. The number of terms is greater by one than the exponent of the binomial. 2. In the first term the exponent of a is the same as the exponent of the binomial, and the exponent of a decreases by one in each succeeding term. 3. b appears in the second term with 1 for an exponent, and its exponent increases by 1 in each succeeding term. 4. The coefficient of the first term is 1. 5. The coefficient of the second term is the same as the exponent of the binomial. 6. The coefficient of each succeeding term is found from the next preceding term by multiplying the coefficient of that term by the exponent of a and dividing the product by a number greater by one than the exponent of b. 220. If b is negative, the terms in which the odd powers of b occur are negative. Thus, 2. 3a2b+3ab2 — b3. (a — b)1 = a1 — 4 a3b + 6 a2b2 -- 4 ab3 + ba. By the above rules any power of a binomial of the form ab may be written at once. NOTE. The double sign + is read plus or minus; and a ±b means a+bora - b. 221. The same method may be employed when the terms of a binomial have coefficients or exponents. 1. Find the third power of 5 x2 — 2 y3. Since (a b)8 = a3 — 3 a2b+ 3 ab2 — b3, putting 5x2 for a, and 2 y3 for b, we have = (5 x2 - 2 y3)3 (5 x2)3 — 3 (5 x2)2 (2 y3) + 3 (5 x2) (2 y3)2 — (2 y3)3 = 125 x 150 x1y3 + 60 x2y — 8 yo. 2. Find the fourth power of x2 y. putting x2 for a, and y for b, we have = (x2)1 — 4 (x2) 3 († y) + 6 (x2)2 († y)2 — 4x2 (†y)3 + (‡y)* 222. In like manner, a polynomial of three or more terms may be raised to any power by enclosing its terms in parentheses, so as to give the expression the form of a binomial. 1. (a+b+c)3 = [a + (b + c)]3 = a3 +3a2 (b+c) + 3 a (b + c)2 + (b + c) 3 = a3 + 3 a2b+3a2c + 3 ab2 + 6 abc +3 ac2+48 + 3 b2c + 3 bc2 + c3. 2. (x3-2x2+3x+4)2 = [(x3 − 2 x2) + (3 x + 4)]2 = (x3 − 2 x2)2 + 2 (x3 − 2 x2) (3 x + 4) + (3 x + 4)2 = x2 - 4x5 +4x+6x1 4x3-16x2+9x2+24x + 16 = x6 — 4x5 +10 x1 - 4x3- 7 x2 + 24 x + 16. 223. The operation of finding any required root of an expression is called Evolution. power cannot be found exactly. A root of an imperfect Thus, the exact value of the square root of 2 can be written only as √2, and the exact value of the cube root of 4 can be written only 3 as V4. Approximate values of these expressions, however, can be found by annexing ciphers and extracting the root. 224. Index Law for Evolution. If m and n are positive Thus, the cube root of a is a = a2; the fourth root of 81 a12 is found by taking the fourth root of 81 and of a12; and is 3 a3. Hence, 225. To Find the Root of a Simple Expression, Take the required root of the numerical coefficient, and divide the exponent of each letter by the index of the required root. 226. From the Law of Signs it is evident that: 1. Any even root of a positive number will have the double sign, . 2. There can be no even root of a negative number. For V- x2 is neither + x nor -x; since the square of + x = + x2, and the square of − x = + x2. The indicated even root of a negative number is called an imaginary number. 3. Any odd root of a number will have the same sign as Any required root of a fraction is found by taking the required root of the numerator and of the denominator. |