118. EVOLUTION. DEFINITION. The root of any proposed expression is that quantity which being multiplied by itself the requisite number of times produces the given expression. The operation of finding the root is called Evolution: it is the reverse of Involution. 119. By the Rule of Signs we see that (1) any even root of a positive quantity may be either positive or negative; (2) no negative quantity can have an even root; (3) every odd root of a quantity has the same sign as the quantity itself. Note. It is especially worthy of remark that every positive quantity has two square roots equal in magnitude, but opposite in sign. Example. In the present chapter, however, we shall confine our attention to the positive root. Examples. √ab1 = a3b2, because (a3b2)2 = aba. -x=-x3, because ( − x3)3 = − x9. 20 c0c4, because (c4)5 = c20. 81x12 = 3x3, because (3x3) = 81x12. 120. From the foregoing examples we may deduce a general rule for extracting any proposed root of a simple expression : Rule. (1) Find the root of the coefficient by Arithmetic, and prefix the proper sign. (2) Divide the exponent of every factor of the expression by the index of the proposed root. EXAMPLES XVI. a. Write down the square root of each of the following expressions : a12 Write down the cube root of each of the following expressions : 14. -a6b3. 11. 16x64 25 12. 4. a6b2c16 -27x9. 64a27b3 17. 18. 19. 20. 27 y15 2766 Write down the value of each of the following expressions : 121. By the formulæ in Art. 115 we are able to write down the square of any binomial. Thus (2x+3y)2=4x2+12xy+9y2. Conversely, by observing the form of the terms of an expression, it may sometimes be recognised as a complete square, and its square root written down at once. Example 1. Find the square root of 2522 - 40xy + 16y2. The expression = (5x)2 - 2. 20xy + (4y)2 = (5x)2 – 2(5x)(4y)+(4y)2 122. When the square root cannot be easily determined by inspection we must have recourse to the rule explained in the next article, which is quite general, and applicable to all cases. But the student is advised, here and elsewhere, to employ methods of inspection in preference to rules. To Find the Square Root of a Compound Expression. 123. Since the square of a+b is a2+2ab+b2, we have to discover a process by which a and b, the terms of the root, can be found when a2+2ab+b2 is given. The first term, a, is the square root of a2. Arrange the terms according to powers of one letter a. The first term is a2, and its square root is a. Set this down as the first term of the required root. Subtract a2 from the given expression and the remainder is 2ab+b2 or (2a+b)×b. Now the first term 2ab of the remainder is the product of 2a and b. Thus to obtain b we divide the first term of the remainder by the double of the term already found; if we add this new term to 2a we obtain the complete divisor 2a+b. The work may be arranged as follows: Example Find the square root of 9x2 – 42xy +49y2. Explanation. The square root of 922 is 3x, and this is the first term of the root. By doubling this we obtain 6x, which is the first term of the divisor. Divide - 42xy, the first term of the remainder, by 6x and we get -7y, the new term in the root, which has to be annexed both to the root and divisor. Next multiply the complete divisor by - 7y and subtract the result from the first remainder. There is now no remainder and the root has been found. 124. The rule can be extended so as to find the square root of any multinomial. The first two terms of the root will be obtained as before. When we have brought down the second remainder, the first part of the new divisor is obtained by doubling the terms of the root already found. We then divide the first term of the remainder by the first term of the new divisor, and set down the result as the next term in the root and in the divisor. We next multiply the complete divisor by the last term of the root and subtract the product from the last remainder. If there is now no remainder the root has been found; if there is a remainder we continue the process. Example. Find the square root of 25x2a2-12xa3 + 16x4+4a4 - 24x3α. Rearrange in descending powers of x. 16x1-24x3a +25x2a2 – 12xa3 +4a4 ( 4x2 - 3xa +2a2 Explanation. When we have obtained two terms in the root, 4x2-3xa, we have a remainder 16x2a2-12xa3 + 4a1. Double the terms of the root already found and place the result, Sx2-6xa, as the first part of the divisor. Divide 16x2a2, the first term of the remainder, by 8x2, the first term of the divisor; we get +2a2 which we annex both to the root and divisor. Now multiply the complete divisor by 2a2 and subtract. There is no remainder and the root is found. 125. Sometimes the following method may be used. Example. Find by inspection the square root of 4a2+b2+c2+4ab-4ac - 2bc. Arrange the terms in descending powers of a, and let the other letters be arranged alphabetically; then the expression = 4a2+4ab - 4ac + b2 - 2bc + c2 = 4a2+4a(bc) + (b − c)2 = · (2a)2+2. 2a(b − c) + (b − c)2 ; whence the square root is 2a +(b−c). [Art. 121.] |