But is is best to set down both the co-efficients and the powers of the letters at once. in one line, without the intermediate lines in the above example, as in the example here below. 2. L.et a - x be involved to the 6th power. The terins with the co-efficients will be 6a5+ 15a*x2 20a3x3 + 15a’x4 6ax: + 36. 3. Required the 4th power of a Ans. a* - 4a3x + ba?r? 4ar' + 3*, And tbus any other powers may be set down at once, in the same manner ; which is the best way. EVOLUTION. EVOLUTION is the reverse of Involution, being the method of finding the square root, cube root, &c. of any giver quantity whether simple or compound. CASE I. To find the Roots of Simple Quantities. EXTRACT the root of the co-efficient for the numeral part ; and divide the index of the letter or letters, by the index of the power, and it will give the root of the literal part ; then annex this to the former, for the whole root sought*. * Any even root of an affirmative quantity, may be either + or - : thus the square root of + a' is either + a, or a; because + ax torta , and -ax-Q + a2 also. But an odd root of any quantity will have the same sign as the quantity itself : thus the cube root of + 23 is ta and the cube root of a3 is - a; for + ax + axtatas, and - a X a X Any even root of a negative quantity is impossible ; for neither tax + a, nor - ax - a can produce - al. Any root of a product, is equal to the like root of each of the factors multiplied together. And for the root of a fraction, take the root of the numerator, and the root of the denominator. EXAMPLES EXAMPLES. 1. The square root of 4uo, is 22. 5,272 592 62 ab 3. The square root of -, or v is N 5. 9c? 9c2 3c 160*66 2ab2 4. The cube root of is 2a. 27c3 3c 5. To find the square rooi of 2a?6*. Ans, aö? v 2. 6. To find the cube root of 642566. Ans. 49b%. 82% 12 2 7 To find the square root of Ans. 2ab V 3c 8. To find the 4th root of 81a*66. Ans. 3ab vb. 9. To find the 5th root of 32a566. Ans 2ab b. 303 CASE IL To find the Square Root of a Compound Quantity. Tais is performed like as in numbers, thus : 1 Range the quantities according to the dimensions of one of the letters, and set the root of the first term in the quotient. 2. Sihtract the square of the root, thus found, from the first term, and bring down the next two terms to the remainder for a dividend ; and take double the root for a divisor. 3 Divide the dividend by the divisor, and annex the re. sul: both to the quotient and to the divisor. 4. Multiply the divisor thus increased, by the term last set in the quotient, and subtract the product from the dividend. And so on, always the same, as in common arithmetic, EXAMPLES. 1. Extract the square root ofe*-4236 + 6a36? -4ab3 + 64. at - 4a36 + 4a3f 4ab3 + 6*(- 2ab + 68 the root, 2. Find the root of a* + 4a3b + 10a3b2 + 12ab3 + 6*. a* + 4036 + 100262 + 12ab3 + 64 ( al + 2ab + 362, 2a2 + 2ab ) 4236 + 10a3b2 4036 + 4a"62 2ao + 4ab+ 362 ) 642 )2 + 12ab3 + 64 ба 262 + 12ab3 + 64 3. To find the square root of a* + 4a3+ 6a? + 4a + 1. Ans. q2 + 2a +1. 4. Estract the square root of a* 2a3 + 2a'att. Ans. r' - + . 5. It is required to find the square root of a? ab. b 6 63 &C. 2 sa 16a? Ans. a CASE III. To find the Roots of any Powers in General. This is also done like the same roots in numbers, thus : Find the root of the first term, and set it in the quotient. Subiract its power from that term, and bring down the second term for a dividend. Involve the rool, last found, to the next lower power, and multiply it by the index of the given power, for a divisor.—Divide the dividend by the di. visor, and set the quotient as the next term of the root. Involve now the whole root to the power to be extracted ; then subtract the power thus arising from the given power, and divide the first term of the remainder by the divisor first found; and so on till the whole is finished*. EXAMPLES. * As this method, in high powers, may be thought too laborious, it will not be improper to observe, that the routs of compound quantities may sometimes be easily discovered, thus : Extract the roots of some of the most simple terms, and connect them together by the sign + or -, as may be judged most suitable for the purpose.--Involve the compound root, thus found, to the proper power; then, if this be the same with the given quantity, it is the root required. - But if it be found to differ only in some of the signs, change them fom * to -, or from - to +, till its power agrees with the given one throughout. Thus EXAMPLES. 1. To find the square root of a*-2a36+3a3b2-%ab% +04. ak-2016 + 3a2b2 - 2ab3 + 6* (al ab +62 2. Find the cube root of a6 625 + 212 44a3 + 63a* 54a + 27. 26-625 +21a*- 44a3 + 63a— 54a + 27 (aa-2a + 3. a6--6.25 +2126-44a3+63a2 – 54a+27 = (a? - 20 - 3)$. Ans. O btx. 3. To find the square root of a2 2ab + 2ax + 62 26x + x2 4. Find the cube root of a6 - 3a5 + 9a" – 1323 + 1822 12a + 8. Ans. a? -Q + 2. 5. Find the 4th root of 81a4- 21623 + 2162262 96263 +1664. Ans. 3a - 26. 6. Find the 5th root of as 100* + 40as 80a2 + 800 32. 2. 7. Required the square root of i r%. 8. Required the cube root of i x3. *Ans, a Thus, in the 5th example, the root 3a - 2b, is the difference of the roots of the first and last terms; and in the third example, the root a-b + x, is the sum of the roots of the 1st, 4th, and 6th terms. The same may also be observed of the 6th example, where the root is found from the first and last terms. SURDS. SURDS. Surds are such quantities as have not exact values in numbers; and are usually expressed by fractional indices, or by means of the radical sign v. Thus, 36, or 3, denotes the square root of 3 ; and or ū22, or 4, the cube root of the square of 2 ; where the numerator shows the power to which the quantity is to be raised, and the denominator its root. 21 PROBLEM I. To Reduce a Rational Quantity to the Form of a Surd. Raise the given quantity to the power denoted by the index of the surd; then over or beiore this new quantiiy set the radical sign, and it will be the form required. EXAMPLES. i. To reduce 4 to the form of the square root. First, 4? = 4 X 4 = 16; then ✓ 16 is the answer. 2. To reduce 3uo to the form of ihe cube root. First 3a2 X 3a? x 3a2 = (322)3 = 27a6 ; then 27a6 or (2729) # is the answer. 3. Reduce 6 to the form of the cube root. Ans. (216)} or 7216. 4. Reduce Jab to the form of the square root. Ans. ✓ Ha?b. 5. Reduce 2 to the form of the 4th root. Ans. (16) 6. Reduce aš to the form of the 5th root. 7. Reduce a + r to the form of the square root. 8. Reduce a I to the form of the cube root. PROBLEM II. To Reduce Quantities to a Common Inder. 1.'Reduce the indices of the given quantities to a common denominator, and involve each of them io the power denored by its numerator; then 1 set over the common denominator will form the common index, Or, |