Question

下列表达式等价于哪一个？ $\begin{aligned} &(2+3)(2^2+3^2)\\ &\quad\cdot(2^4+3^4)(2^8+3^8)\\ &\quad\cdot(2^{16}+3^{16})(2^{32}+3^{32})\\ &\quad\cdot(2^{64}+3^{64})? \end{aligned}$

Which of the following is equivalent to $\begin{aligned} &(2+3)(2^2+3^2)\\ &\quad\cdot(2^4+3^4)(2^8+3^8)\\ &\quad\cdot(2^{16}+3^{16})(2^{32}+3^{32})\\ &\quad\cdot(2^{64}+3^{64})? \end{aligned}$

$3^{127} + 2^{127}$

$3^{127} + 2^{127} + 2 \cdot 3^{63}$ $+ 3 \cdot 2^{63}$

$3^{128} - 2^{128}$

$3^{128} + 2^{128}$

$5^{127}$

答案：C

知识点：平方差裂项相消

难度评级：1070

视频讲解：

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文字解答：

注意，乘以 $3-2=1$ 不会改变原式的值，并且会产生一连串平方差。 $3^{128}-2^{128}$ 。 $\begin{aligned} (3-2)(3+2)&=3^2-2^2,\\ (3^2-2^2)(3^2+2^2)&=3^4-2^4, \end{aligned}$

这样一路望远镜相乘，最终得到。所以正确答案是 C。

Multiply the product by $3-2=1.$ Repeatedly applying the difference-of-squares identity gives $\begin{aligned} (3-2)(3+2)&=3^2-2^2,\\ (3^2-2^2)(3^2+2^2)&=3^4-2^4, \end{aligned}$ and the same cancellation continues through the final factor. Therefore the product is $3^{128}-2^{128}.$

Thus, C is the correct answer.

2021 AMC 10A Spring 第 10 题

答案：C

难度评级：1070

视频讲解：

文字解答：

其他年份的第 10 题