高一对数运算：计算lg^3(2)+lg^3(5)+3lg(2)lg

令
X=lg(2)
Y=lg(5)

lg^3(2) + lg^3(5))+ 3lg(2)lg(5)
= X^3 + Y^3 + 3XY
= (X+Y) (X^2 -XY + Y^2) + 3XY

一. X+Y = log(2) + log(5) = log(2*5) = log(10) = 1

所以,

lg^3(2) + lg^3(5))+ 3lg(2)lg(5)
= (X+Y) (X^2 -XY + Y^2) + 3XY
= 1 * (X^2 -XY + Y^2) + 3XY
= X^2 +2XY + Y^2
= (X+Y)^2
= 1^2
= 1

你究竟想表达什么啊，看不出来！！

令a=lg2,b=lg5,则a+b=lg10=1

lg^3(2)+lg^3(5)+3lg(2)lg(5)
=a^3+b^3+3*a*b
=(a+b)(a^2-ab+b^2)+3ab
=a^2-ab+b^2+3ab
=a^2+2ab+b^2
=(a+b)^2
=1