To extract the square root of a compound quantity. RULE 1.—Range the terms, of which the quantity is composed, according to the dimensions of some letter in them, beginning with the highest, 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 two next terms to the remainder for a dividend. 3. Divide the dividend, thus found, by double that part of the root already determined, and set the result both in the quotient and divisor. 4. Multiply the divisor, so increased, by the term of the root last placed in the quotient, and subtract the product from the dividend; and so on, as in common arithmetic. EXAMPLES. 1. Extract the square root of x1 4x3+6x2 4x + 1. 4x+6x-4x+1(x2-2x+1 2x + 1, the root required. 2. Extract the square root of 4a2 + 12a3x + 13a2x2 + ax36 4a* + 12a3x + 13a2x2 + 6ax3 + x2 (2a2 + 3ax + x2 Note. When the quantity to be extracted has no exact root, the operation may be carried on as far as is thought necessary, or till the regularity of the terms show the law by which the series would be continued. EXAMPLE. 1. It is required to extract the square root of 1+x. 203 5x4 + &c. 8 Here, if the numerators and denominators of the two last terms be each multiplied by 3, which will not alter their values, the root will become 3x3 + 3.5x4 3.5.7x5 + &c. 2 2.4 2.4.6 2.4.6.8 2.4.6.8.10' where the law of the series is manifest. EXAMPLES FOR PRACTICE. 2. It is required to find the square root of aa + 4a3x + 6a2x2+4ax3 + x2. Ans. a2+2ax + x2. 3. It is required to find the square root of x1- 2x3 + 4. It is required to find the square root of 4x 12x3x2. 6x+9. 5. Required the square root of x6 + 4x5 +10x2 + 20x3 + 25x2+24x + 16. Ans. x32x2 + 3x + 4. 6. It is required to extract the square root of a2 + b. 7. It is required to extract the square root of 2, or of 1 +1. Ans. 1+ + 16-32 + 64, &c. — CASE III. 8 To find any root of a compound quantity. 1 RULE. Find the root of the first term, which place in the quotient; and having subtracted its corresponding power from that term, bring down the second term for a dividend. Divide this by twice the part of the root above determined, for the square root; by three times the square of it, for the cube root, and so on; and the quotient will be the next term of the root. Involve the whole of the root, thus found, to its proper power, which subtract from the given quantity, and divide the first term of the remainder by the same divisor as before; and proceed in this manner till the whole is finished.* EXAMPLES. 1. Required the square root of a 2a3x + 3a2x2 aa aa — 2α3x + 3a2x2 -- 2ax3 + x1 (a2 — ax + x2 28x3 * As this rule, in high powers, is often found to be very laborious, it may be proper to observe, that the roots of various compound quantities may sometimes be easily discovered, as follows: Extract the roots of all the simple terms, and connect them together by the signs or, as may be judged most suitable for the purpose; then involve the compound root, thus found, to its proper power, and if it be the same with the given quantity, it is the root required. But if it 3. Required the square root of 4a2 12ax +9x2. Ans. 2a 3x. 4. Required the square root of a2+2ab+2ac + b2 + 2bc Ans. a+b+c. 6x5+15x4 20x3+ Ans. x3 2x + 1. 96α3x +216a2x2 Ans. 2a. 3x. 80x80x3 40x2+ Ans. 2x 1. OF IRRATIONAL QUANTITIES, OR SURDS. IRRATIONAL Quantities or Surds, are those of which the values cannot be accurately expressed in numbers; and are usually expressed by means of the radical sign √, or by fractional indices; in which latter case, the numerator shows the power the quantity is to be raised to, and the denominator its root. a3 the cube root of the square of a, &c.* be found to differ only in some of the signs, change them from+to—, or from to, till its power agrees with the given one throughout. Thus, in the third example next following, the root is 2a-3x, which is the difference of the roots of the first and last terms; and in the fourth example, the root is a+b+c, which is the sum of the roots of the first, fourth, and sixth terms. The same may also be observed of the sixth example, where the root is found from the first and last terms. * A quantity of the kind here mentioned, as for instance 2, is called an irrational number, or a surd, because no number, either whole or fractional, can be found, which, when multiplied by itself, will produce 2. But its approximate value may be determined to any degree of exactness, by the common rule for extracting the square root, being 1 and certain non-periodic decimals, which never terminate. CASE I. To reduce a rational quantity to the form of a surd. RULE.-Raise the quantity to a power corresponding with that denoted by the index of the surd; and over this new quantity place the radical sign, or proper index, and it will be of the form required. EXAMPLES. 1. Let 3 be reduced to the form of the square root. 2. Reduce 2x2 to the form of the cube root. Ans. 1 Here (2x2)3 = 8x; whence / 8xo, or (8x®)3. 3. Let 5 be reduced to the form of the square root. 4. Let 5. Let 6. Let a2 √a + √b, Ans. √(25). 3x be reduced to the form of the cube root. Ans. (273). 3/ 2a be reduced to the form of the fourth root. Ans. (16a). / be reduced to the form of the fifth root, and √ a 2a 5 and a b√ a to the form of the square root. a Ans. 3/ a1o, √ (a + 2 √ ab + b), √(a), and ✓ b2 Note. Any rational quantity may be reduced by the above rule, to the form of the surd to which it is joined, and their product be then placed under the same index or radical sign. EXAMPLES. да Thus 2√2 √4X √2 = √(4 x 2) = √8 And 23/4=8XV4 = (8 X 4) = 3/32 / Also 3 √α = √ (9 X a) = √9a And 4a VXV4a= (X 4a) = V 1. Let 5 √6 be reduced to a simple radical form. 2. Let 5a be reduced to a simple radical form. To reduce quantities of different indices, to others that shall have a given index. RULE.-Divide the indices of the proposed quantities by |