CHAPTER XIII POWERS AND ROOTS SECTION 1, POWERS. SECTION 2, SQUARE ROOT. SECTION 3, CUBE ROOT. SECTION 4, OTHER ROOTS. § 1. POWERS 190. Power of a Monomial. In Chapter XI the rule was given that a monomial may be raised to any power by multiplying each exponent of its factors by the exponent of the power. The reason for the rule will now be developed. In (4a2b3)3 the exponent 3 denotes that 4a2b3 is to be taken how many times as a factor? Therefore 4 must be used how many times as a factor? And 63 how many times? Therefore the cube of 4a2b3 equals what power of 4 times what power of a times what power of b? Could these powers of 4, a, and b, have been written directly by multiplying the exponents of these three factors by 3, the exponent of the power? Therefore (4a2b3)3 equals what? In order, therefore, to raise a monomial to any power what operation should be performed (1) on the numerical coefficient? (2) on the exponents of its literal factors? 191. The Binomial Formula. As shown in the preceding chapter a binomial may be raised to any power by substitution in the binomial formula. By the use of marks of parenthesis this formula may also be used for determining any power of a trinomial and an expression having four terms. and Thus (a+b+c)3=(a+b+c)3, (a+b+c+d) 4 =(a+b+c+d) 4; the vinculum being used in both illustrations to transform the given expressions into binomials. In this connection it should not be forgotten that the square of any polynomial is most readily obtained by the law given in Chapter V. 192. Examples. By inspection expand the following. In each example state what law was applied. 193. Any Root of a Monomial. A monomial may be raised to any power by what operation on (1) its numerical coefficient, (2) the exponents of its literal factors? The extraction of a root is the inverse of the determination of a power. Therefore any root of a monomial may be extracted by what operation on and (1) its numerical coefficient, (2) the exponents of its literal factors? Thus V8x2y9 equals what? 16a2b-4c equals what? Reduce each of the two results to a form having no fractional and no negative exponents. 194. Sign of the Root. Perform the following indicated operations: (2x2y)2, (-2x2y)2, (−2x2y)1. (2x2y)3, (-2x2y)3, (-2x2y)5. The results show that: (1) Even powers have what sign? (2) Odd powers have what sign? (3) Even roots of positive quantities have what sign? (4) Even roots of negative quantities are impossible. Why? (5) Odd roots have what sign? 195. Examples. Extract the indicated roots of the following and reduce the results to forms with no fractional and no negative exponents: 196. Square Root of a Polynomial. The following instructions should enable you to extract the square root of any polynomial. (1) Arrange the terms in descending or ascending order with respect to the same letter. (2) Enter at the right the square root of the first term of the polynomial. (3) Subtract its square from the first term and bring down the next term of the polynomial. (4) At the left set down a trial divisor determined from the following law: Trial divisor = 2X root. (5) Enter in the root the number of times the trial divisor is contained in the first term. (6) Add to the trial divisor to form the complete divisor, last term of root. |