Having arranged the terms according, to the dimensions of one letter, a, the square root of the first term a' is a, the first factor in the root; subtract its square from the whole quantity, and bring down the remainder 2 a 6-14' ; divide 2 a b by 2 a, and the result is b, the other factor in the root ; then multiply the sum of twice the first factor and the second (2a-1-6), by the second (b`„ and subtract this pro. duct (2 a bl-b.) from the remainder. If there be no more terms, consider a-4-6 as a new value of a; and the square, that is rP-1-2 a bi-61, having, by the first part of the process, been subtracted from the proposed quantity, divide the remainder by the double of this new value of a, for a new factor in the root; and for a new subtrahend, multiply this factor by twice the sum of the former factors increased by this factor. The process must be re peated till the root, or the neoessary ap proximation to the root, is obtained.
Ex. 1. To extract the square root of 0+2 a a c+2 6 c-}-c'.
a'4-2 a b+61-1-2 a c-I-2 6 (a-}-6-1-r a' 2 a+6)2 a b.4-b= 2 a b-}-b' 2 a + 2 6+ cy • 2 a e-1-2 c+c= 2 a c+2 b el-cz • • • Ex. 2. To extract the square root of at x2 a x± a X 4 as X x.
xs —ct — • • Ex. 3. To Extract the square root of / +x.
T-8 1 2 , x2 x 4 — 8 4 X3 x4 • 4 8 64 8 64 It appears from the second example, that a trinomial a a x + in which four times the product of the first and last terms is equal to the square of the middle term, and a complete square, or x2 42 Xx2.
4 The method of extracting the cube root is discovered in the same manner. a3+3 a2 6+3 a b2-143 (a+ b a3 3 a2)3 a2 b-1-3 a 62 +63 3 ,12 b+3 a b2+63 • The cube root of a3+3 6+3 a ba_f.63 is a+b ;- and to obtain a-f-b from this compound quantity, arrange the terms as before, and the cube root of the first term, a3, is a, the first factor in the root ; sub tract its cube from the whole quantity, and divide the first term of the remainder by3 a 2, the result is b, the second factor in the root; then subtract 3 n2 6+3 a b2+b3 from the remainder, and the whole cube of a±b has been subtracted. If any quantity be left, proceed with a.-1-6 as a new a, and divide the last remainder by 3 . 2 for a third factor in the root ; and thus any number of factors may be obtained.