divide the first term of the remainder by the same divisor as before ; and proceed in this manner till the whole is. finished. (p) (p). 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 -H 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 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 ma lso be observed of the sixth example, where the root is foun from the first and last terms. IRRATIONAL quantities, or surds, are such as have no exact root, being 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. Thus, V2, or 23, denotes the square root of 2; and 3/a”, or aff, is the square of the cube root of a, &c. (q) CASE I. To reduce a rational quantity to the form of a surd. RULE. w 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. Here 3×3=3°=9 ; whence v.9 Ans. * 2. Reduce 2x2 to the form of the cube root. Here (2x2)*=8x"; whence 3/82°, or (82%); Ans. 3. Let 5 be reduced to the form of the square root. 4. Let —3r be reduced to the form of the cube root. 5. Let –2a be reduced to the form of the fourth root. 6. Let a? be reduced to the form of the fifth root, and va-H vb, #and o: to the form of the square root. 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. (7) A quantity of the kind here mentioned, as for instance v. 2, is callet; an irrational number, or a sorrd, because no number, either whole or fractional, can be found, which when multipled 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 deci: mals, which never terminate. Examples. 3. Let * vi. 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 the given index, and the quotients will be the new indices for those quantities. Then, over the said quantities, with their new indices, place the given index, and they will be the equivalent quantities required. EXAMPLES. 1. Reduce 33 and 2% to quantities that shall have the index+. tities required. -l 2. Reduce 54 and 63 to quantities that shall have the common index & 3. Reduce 23 and 48 to quantities that shall have the - 1 common index l 8. w 4. Reduce as and do to quantities that shall have the common index w 4 5. Reduce a} and b to quantities that shall have the - I Common index 5 * * JNote. Surds may also be brought to a common index, by reducing the indices of the quantities to a common denominator, and then involving each of them to the power denoted by its numerator. 3. Reduce ał and as to quantities that shall have a common index. l 4. Reduce a 3 and bi to quantities that shall have a Qommon index. l L 5. Reduce an and b” to quantities that shall have a common index. |