Question

三个数 $10^8$ 、 $5^{12}$ 和 $2^{24}$ 的正确大小顺序是什么？

What is the correct ordering of the three numbers, $10^8,$ $5^{12},$ and $2^{24}?$

2^{24} < 10^8 < 5^{12}

2^{24} < 5^{12} < 10^8

5^{12} < 2^{24} < 10^8

10^8 < 5^{12} < 2^{24}

10^8 < 2^{24} < 5^{12}

知识点：指数

比较 $2^{24}$ 、 $10^8=(2\cdot5)^8$ 和 $5^{12}$

Compare $2^{24}$ , $10^8=(2\cdot5)^8$ , and $5^{12}$ .

把幂改写成带有可比较的八次方因子的形式

Rewrite the powers so they have comparable eighth-power factors.

首先， $2^{24} = (2^8)(4^8) < (2^8)(5^8) = 10^8\text{。}$

接着， $10^8 = (2^8)(5^8) =$ $(4^4)(5^8) < (5^4)(5^8) = 5^{12}\text{。}$ 因此 $2^{24} < 10^8 < 5^{12}$ 。

所以正确答案是 A。

First, we get $2^{24} = (2^8)(4^8) < (2^8)(5^8) = 10^8.$

Next, we get $10^8 = (2^8)(5^8) =$ $(4^4)(5^8) < (5^4)(5^8) = 5^{12}.$ This means $2^{24} < 10^8 < 5^{12} .$

Thus, the answer is A .