AMC8 2016

AMC8 2016 · Q7

AMC8 2016 · Q7. It mainly tests Exponents & radicals, Perfect squares & cubes.

Which of the following numbers is not a perfect square?

下列哪个数不是完全平方数？

(A) $1^{2016}$ $1^{2016}$

(B) $2^{2017}$ $2^{2017}$

(D) $4^{2019}$ $4^{2019}$

(E) $5^{2020}$ $5^{2020}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

The numbers $1^{2016}$, $3^{2018}$, $5^{2020}$ have even exponents and hence are squares. The number $2^{2017}$ is not a perfect square because it is twice a square $2(2^{1008})^2$. Since $4^{2019} = (2^{2})^{2019} = 2^{4038}$, it is also a perfect square.

数 $1^{2016}$、$3^{2018}$ 和 $5^{2020}$ 的指数为偶数，因此它们都是完全平方数。数 $2^{2017}$ 不是完全平方数，因为它等于一个平方数的两倍：$2(2^{1008})^2$。由于 $4^{2019}=(2^{2})^{2019}=2^{4038}$，它也是一个完全平方数。

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