1990 AMC 12 第 22 题

Question

若方程

x^6=-64

的六个解写成

a+bi

的形式，其中

a

和

b

都是实数，那么所有满足

a\gt0

的解的乘积为

If the six solutions of

x^6=-64

are written in the form

a+bi,

where

a

and

b

are real, then the product of those solutions with

a\gt0

is

-2

0

2i

4

16

将

-64

写成极形式，并列出它的六个六次方根

Write

-64

in polar form and list its six sixth roots

实部为正的两个根构成一对共轭复数

The roots with positive real part form a conjugate pair

这些根的模为

2

，辐角为

\frac\pi6+\frac{k\pi}{3},\qquad k=0,1,\ldots,5\text{。}

实部为正的两个根的辐角分别为

\frac{\pi}{6}

和

\frac{11\pi}{6}

。它们互为共轭，模均为

2

，所以乘积为

2^2=4

。

所以正确答案是 D。

The roots have modulus

2

and arguments

\frac\pi6+\frac{k\pi}{3},\qquad k=0,1,\ldots,5.

The roots with positive real part have arguments

\frac{\pi}{6}

and

\frac{11\pi}{6}.

They are conjugates of modulus

2,

so their product is

2^2=4.

Thus the correct answer is D.

其他年份的第 22 题