NOVELTY - Manufacture of positive electrode active material involves wet-mixing nickel-cobalt-manganese composite hydroxide particles obtained by crystallization and lithium compound, drying obtained slurry, baking obtained powder at 800-1000 degrees C in an oxidative atmosphere, and washing obtained hexagonal lithium-nickel-cobalt-manganese complex oxide particles having layered structure. The hexagonal lithium-nickel-cobalt-manganese complex oxide particles have average particle size of 3-8 mu m. The nickel-cobalt-manganese composite hydroxide particles have average particle size of 3-7 mu m.    USE - Manufacture of positive electrode active material used for nonaqueous electrolyte secondary battery (all claimed) e.g. lithium ion secondary battery used in power supply devices e.g. electric vehicle e.g. hybrid motor vehicle, portable electronic device e.g. mobile telephone and notebook computer, and power tool.    ADVANTAGE - The method enables manufacture of positive electrode active material with high productivity. The nonaqueous electrolyte secondary battery containing positive electrode active material has high output, cycle characteristics and capacity ands reduced positive electrode resistance.    DETAILED DESCRIPTION - Manufacture of positive electrode active material involves wet-mixing nickel-cobalt-manganese composite hydroxide particles obtained by crystallization and lithium compound, drying obtained slurry, baking obtained powder at 800-1000 degrees C in an oxidative atmosphere, and washing obtained hexagonal lithium-nickel-cobalt-manganese complex oxide particles having layered structure with water. The hexagonal lithium-nickel-cobalt-manganese complex oxide particles are of formula: Li1+uNiaCobMncMdO2, and have average particle size of 3-8 mu m. The hexagonal lithium-nickel-cobalt-manganese complex oxide particles have index which is breadth of particle size distribution ((d90-d10)/MV) of 0.60 or less and ((d10)/MV) of 1.30 or less. The nickel-cobalt-manganese composite hydroxide particles are of formula: NiaCobMncMd(OH)2+a', and have average particle size of 3-7 mu m. The nickel-cobalt-manganese composite hydroxide particles have index which is breadth of particle size distribution ((d90-d10)/MV) of 0.55 or less.    a+b+c+d=1;    a=0.3-0.8;    b=0.1-0.4;    c=0.1-0.4;    d=0-0.1;    a'=0-0.5;    M=Mg, Ca, Al, Ti, V, Cr, Zr, Nb, Mo or W;and    u=-0.05 to 0.15.    An INDEPENDENT CLAIM is included for nonaqueous electrolyte secondary battery.    DESCRIPTION OF DRAWING(S) - The drawing shows a flowchart explaining manufacture of positive electrode active material. (Drawing includes non-English language text)