EVOLUTION. 185
' ?/3 2/2 y X X^ X^ r^ r2 18 27 27 27 3 X a:2 x^ „ x3 12x2 , 54 X ,io , 108 a 48 a^ 8 ^3 a^ a^ a x x^ x^ g 64 gs 192 a2 240 a ^q^ ^ 60 x 12 x^ ^ x^ x-^ x^ X a «2 (jj3 9 6 6 6a ^ a3 3^2 3 52 53 a & 6^ &2 (^2 (1(3 10 ^^ - ^Q^^ _ 90x2 , 8x^ , 108 X _ 27 j_ 48x_^ y yS y2 yQ y yb
203. Some Higher Roots. The fourth root of an expression is obtained by extracting the square root of the square root of the expression.
Similarly by successive applications of the rule for finding the square root, we may find the eighth, sixteenth, ... root. The sixth root of an expression is found by taking the cube root of the square root, or the square root of the cube root.
Similarly by combining the two processes for extraction of cube and square roots, other higher roots may be obtained.
Ex. 1. Find the fourth root of
81 x4 - 216 x3y + 216 x2y2 - 96 xy3 + 16 y4
Extracting the square root by the rule we obtain 9 x2 — 12 xy + 4y2; and by inspection, the square root of this is 3x — 2y, which is the required fourth root.
Ex. 2. Find the sixth root of
By inspection, the square root of this is (x3 - 1/x3)-3(x-1/x), {mmf}} which may be written x3 — 3x + 3/x - 1 x3; and the cube root of this is x — 1/x which is the required sixth root.
