1993 AMC 12 第 6 题

Question

\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=

\sqrt2

16

32

12^{\frac{2}{3}}

512.5

把每个幂都改写成以

2

为底。

Rewrite every power with base

2

从分子和分母中分别提出较小的幂。

Factor the smaller power from the numerator and denominator

将各项都写成

2

的幂，得到

\begin{aligned} \frac{8^{10}+4^{10}}{8^4+4^{11}} &=\frac{2^{30}+2^{20}}{2^{12}+2^{22}}\\ &=\frac{2^{20}(2^{10}+1)} {2^{12}(1+2^{10})}\\ &=2^8 \end{aligned}\text{。}

它的正平方根是

2^4=16

。因此正确答案是 B。

Writing all terms as powers of

2,

\begin{aligned} \frac{8^{10}+4^{10}}{8^4+4^{11}} &=\frac{2^{30}+2^{20}}{2^{12}+2^{22}}\\ &=\frac{2^{20}(2^{10}+1)} {2^{12}(1+2^{10})}\\ &=2^8. \end{aligned}

Its positive square root is

2^4=16.

Thus the correct answer is B.

其他年份的第 6 题