1956 AMC 12 第 9 题

Question

化简

\left[\sqrt[3]{\sqrt[6]{a^9}}\right]^4\left[\sqrt[6]{\sqrt[3]{a^9}}\right]^4

；结果为：

Simplify

\left[\sqrt[3]{\sqrt[6]{a^9}}\right]^4\left[\sqrt[6]{\sqrt[3]{a^9}}\right]^4;

the result is:

a^{16}

a^{12}

a^8

a^4

a^2

将每个

n

次根改写为指数

\frac{1}{n}

Replace each

n

th root by an exponent of

\frac{1}{n}

每个括号内底数的指数为

9\cdot\frac16\cdot\frac13\cdot4

Each bracket has exponent

9\cdot\frac16\cdot\frac13\cdot4

对两个因式中的每一个，嵌套根式和外部四次方将

a

的指数乘以

9\cdot\frac16\cdot\frac13\cdot4=2\text{。}

因此每个因式都是

a^2

，两者之积为

a^2a^2=a^4

。

因此，正确答案是 D。

For each of the two factors, the nested roots and outer fourth power multiply the exponent of

a

by

9\cdot\frac16\cdot\frac13\cdot4=2.

Thus each factor is

a^2,

and their product is

a^2a^2=a^4.

Therefore, the correct answer is D.

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