Roots of Monomials. 194. According to Art. 184, in order to raise a inonomial to any power, we raise the numerical coefficient to the required power, and multiply the exponent of each of the letters by the exponent of the power required. Hence, conversely, to exextract the root of the nu tract any root of a monomial, we merical coefficient, and divide the exponent of each letter by the index of the required root. Thus the cube root of 64ab3 is 4a2b. 195. Sign of the Root.-We have seen, Art. 185, that all powers of a positive quantity are positive; but the even powers of a negative quantity are positive, while the odd powers are negative. Thus +a, when raised to different powers in succession, will give +a, +a2, +a3, +aa, +a3, +ao, +a”, etc. anda, in like manner, will give —a, +a2, -—a3, +aa, —a3, +ao, -a, etc. Hence it appears that if the root to be extracted be expressed by an odd number, the sign of the root will be the same as the sign of the proposed quantity. Thus, Va3——a; and V+a3 =+α. If the root to be extracted be expressed by an even number, and the quantity proposed be positive, the root may be either positive or negative. Thus, va2=±a. If the root proposed to be extracted be expressed by an even number, and the sign of the proposed quantity be negative, the root can not be extracted, because no quantity raised to an even power can produce a negative result. 196. Hence, to extract any root of a monomial, we have the following RULE. 1st. Extract the required root of the numerical coefficient. 2d. Divide the exponent of each literal factor by the index of the required root. 3d. Every even root of a positive quantity must have the double sign, and every odd root of any quantity must have the same sign as that quantity. From Art. 186, it is obvious that to extract any root of a fraction, we must divide the root of the numerator by the root of the denominator. Thus, 15. Find the square root of 64a-2b-4x4. Ans. ±8a-1b-2x2. 16. Find the cube root of -512a-36-6x3. Ans. -2a-2b-3x. Ans. ±(a—b)x3. Square Root of Polynomials. 197. In order to discover a rule for extracting the square root of a polynomial, let us consider the square of a+b, which is a2+2ab+b2. If we arrange the terms of the square according to the dimensions of one letter, a, the first term will be the square of the first term of the root; and since, in the present case, the first term of the square is a2, the first term of the root must be a. Having found the first term of the root, we must consider the rest of the square, namely, 2ab+b2, to see how we can derive from it the second term of the root. Now this remainder may be put under the form (2a+b)b; whence it appears that we shall find the second term of the root if we divide the remainder by 2a+b. The first part of this divisor, 2a, is double of the first term already determined; the second part, b, is yet unknown, and it is necessary at present to leave its place empty. Nevertheless, we may commence the division, employing only the term 2a; but as soon as the quotient is found, which in the present case is b, we must put it in the vacant place, and thus render the divisor complete. The whole process, therefore, may be represented as follows: a2+2ab+b2(a+b a2 2a+b)2ab+b2 If the square contained additional terms, we might continue the process in a similar manner. We may represent the first two terms of the root, a+b, by a single letter, m, and the remaining terms by c. The square of m+c will be m2+2mc+c2. The square of the first two terms has already been subtracted from the given polynomial. If we divide the remainder by 2m as a partial divisor, we shall obtain c, which we place in the root, and also at the right of 2m, to complete the divisor. We then multiply the complete divisor by c, and subtract the product from the dividend, and thus we continue until all the terms of the root have been obtained. 198. Hence we derive the following RULE. 1st. Arrange the terms according to the powers of some one letter; take the square root of the first term for the first term of the required root, and subtract its square from the given polynomial. 2d. Divide the first term of the remainder by twice the root already found, and annex the result both to the root and the divisor. Multiply the divisor thus completed by the last term of the root, and subtract the product from the last remainder. 3d. Double the entire root already found for a second divisor. Divide the first term of the last remainder by the first term of the second divisor for the third term of the root, and annex the result both to the root and to the second divisor, and proceed as before until all the terms of the root have been obtained. If the given polynomial be an exact square, we shall at last find a remainder equal to zero. EXAMPLES. 1. Extract the square root of a1-2a3x+3a2x2-2αx3+x*. a1-2a3x+3a2x2-2ax3+x1 (a2—ax+x2 a1 2a2-ax)-2a3x+3a2x2 -2a3x+ a2x2 2a2-2αx+x2) 2a2x2-2ax3+24 2a2x2-2ax3+x+ For verification, multiply the root a2-ax+x2 by itself, and we shall obtain the original polynomial. 2. Extract the square root of a2+2ab+2ac+b2+2bc+c2. 3. Extract the square root of 10x-10x3-12x5+5x2+9x6-2x+1. 4. Extract the square root of 8αx3+4a2x2+4x2+1662x2+16b++16ab2x. Ans. 2x2+2ax+462. 5. Extract the square root of 15a b2+a6-6a5b-20a3b3+b6+15a2b1-6abs. 6. Extract the square root of 8ab3+a*—4a3b+4l1. 199. When a Trinomial is a Perfect Square.-The square of a+b is a2+2ab+b2, and the square of a-b is a2-2ab+b2. Hence the square root of a2±2ab+b2 is a±b; that is, a trinomial is a perfect square when two of its terms are squares, and the third is the double product of the roots of these squares. Whenever, therefore, we meet with a quantity of this description, we may know that its square root is a binomial; and the root may be found by extracting the roots of the two terms which are complete squares, and connecting them by the sign of the other term. EXAMPLES. 1. Find the square root of 4a2+12ab+9b2. Ans. 2a+3b. 2. Find the square root of 9a2-24ab+1662. 3. Find the square root of 9a-30a3b+25a2b2. 4. Find the square root of 4a2+14ab+16b2, if possible. No algebraic binomial can be a perfect square, for the square of a monomial is a monomial, and the square of a binomial necessarily consists of three distinct terms. |