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. Let a - be involved to the 6th power. The terms with the co-efficients will be a — 6α3x + 15a⭑x2 20a3x3+15a2x2 3. Required the 4th power of a-x. Ans. a -4a3x+6a2x2 6ax + x. And thus 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 given 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 + a2 is either+a, or - a; because + ax + a= +a and ·ax — a + a2 also. But an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of a3 is a and the cube root of 23 is — a ; for + a × + ax+ a = + a3, and a x Any even root of a negative quantity is impossible; for neither +ax+a, nor — -a can produce -ax a2. 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. To find the Square Root of a Compound Quantity. THIS 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. Subtract 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 result 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 of a⭑-4a3b + 6a2b2 —4ab3 +64. 2. Find the root of a⭑+ 4a3b + 10a2b2 + 12ab3 + b♦. 2a2+2ab) 4a3b + 10ab2 4a3b + 4ab2 2 2a2 + 4ab + 3bo ) 6a2¿a + 12ab3 764 6a22+12ab3 + b4 3. To find the square root of a + 4a3*+ 6a2 + 4a + 1. Ans. a*+2a + 1. Ans. 2. -x 5. It is required to find the square root of a2 + 2. b 63 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. -Subtract its power from that term, and bring down the second term for a dividend-Involve the root, 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 divisor, 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 roots 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 from to, or from with the given one throughout. to, till its power agrees Thus EXAMPLES. 1. To find the square root of a⭑—2a3b+3a2b2 — 2ab3+ba. a—2a3b+3a2b2 ·2ab3b4 (a2 ab + b2 -2a3b + 3a2b2—2ab3 + b* = (a2 —ab + b2)2. 2. Find the cube root of a6 6a5+21a 54a + 27 ( a2 —2a + 3. 6a5+12a-8a3 = (a2 - 2a)3 3a) +12a4 a®-6x+21a-44a3+63a2-54a+27= (a2 - 2a-3)3. 3. To find the square root of a2 2bx + x2. 2ab+2ax + b2 Ans. a- b+x. 4. Find the cube root of a6 -3a3 + 9aa 12a +8. 13a3+18a2 Ans. a2-a + 2. 5. Find the 4th root of 81a4-216a3b+ 216a2b2 +166*. 6. Find the 5th root of as - 32. 7. Required the square root of 1 2. 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 bx, 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 ✔. Thus, 34, or ✔ 3, denotes the and 23 square root of 3; or 22, 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. 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 before this new quantity set the radical sign, and it will be the form required. EXAMPLES. i. To reduce 4 to the form of the square root. 13 then 27a or (27a6) is the answer. 3. Reduce 6 to the form of the cube root. Ans. (216) or 216. 4. Reduce lab to the form of the square root. 5. Reduce 2 to the form of the 4th root. 6. Reduce as to the form of the 5th root. Ans. ✔fa2b9. 7. Reduce a +x to the form of the square root. $. Reduce a to the form of the cube root. PROBLEM II. To Reduce Quantities to a Common Index. 1. REDUCE the indices of the given quantities to a common denominator, and involve each of them to the power denoted by its numerator; then 1 set over the common denominator will form the common index, Or, 2. If |