Question

已知a=3√4+3√2+3√1,（3是根指数）那么，3/a+3/a^2+1/a^3的值为______.

Answer 1

已知 a=三次根号4+三次根号2+1.则3/a+3/a^2+1/a³=____
a = 4^(1/3) + 2^(1/3) + 1
令 x = 2^(1/3)
则 a = x² + x + 1
a (x - 1) = (x - 1)(x² + x + 1) = x³ - 1 = 2 - 1 = 1
所以：x = 1 + 1/a
3/a + 3/a² + 1/a³ = (1 + 1/a)³ - 1 ---------- 利用了： (1 + x)³ = 1 + 3x + 3x² + x³
= x³ - 1
= 2 - 1
= 1
.则 3/a + 3/a² + 1/a³ = 1

Answer 2

令x=3√2
a=x²+x+1
(x-1)(x²+x+1)=x³-1=2-1=1
1/a=x-1
原式=3(x-1)+3(x-1)²+(x-1)³
=3x-3+3x²-6x+3+x³-3x²+3x-1
=x³-1
=1