186 ALGEBRA. We conclude the subject of higher roots by giving a rule, which depends upon the Binomial Theorem, for finding the nth root of any multinomial.
(1) Arrange the terms according to the descending powers of some letter. (2) Take the nth root of the first term, and this will be the first term of the root. (3) When any number of terms of the root have been found, subtract from the given multinomial the nth power of the part of the root already found, and divide the first term of the remainder by n times the (n — 1)th power of the first term of the root, and this will be the next term of the root.
204. When an expression is not an exact square or cube, we may perform the process of evolution, and obtain as many terms of the root as we please.
Ex. To find four terms of the square root of 1 + 2x - 2 x2.
1 + 2 a: - 2 ^2(1 + X - f x2 + f x3 1 2 + a; 2 + 2x- 2 ic - 2 x2 2x+ ic2 3x2 3 x2 - 3 x3 + 2 + 2ic- 3^2 + 1x3 3x3-f.x4 3x3 + 3x4
Thus the required result is 1 + x — 3 2 x2 + 3 2 x3.
EXAMPLES XXI. f.
Find the fourth roots of the following expressions:
1. x4 - 28 x3 + 294x2- 1372 X + 240L 2. 16-32+24 8 ^J_ m m'^ m^ m* 3. a* + 8 a^x + 16 x* + 32 ax^ + 24 a^x^. 4. 1 + 4 X + 2 x2 - 8 x3 - 5 x* + Sx-"^ + 2 x« - 4 x" + x8. 5. 1 + 8x + 20x2 + 8x3 -2Gx*-8x5 + 2Ux'i - Sx^ + x«
