AMC8 2010

AMC8 2010 · Q24

AMC8 2010 · Q24. It mainly tests Exponents & radicals.

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

三个数 $10^8$、$5^{12}$ 和 $2^{24}$ 正确的顺序是？

(A) $2^{24} < 10^8 < 5^{12}$ $2^{24} < 10^8 < 5^{12}$

(B) $2^{24} < 5^{12} < 10^8$ $2^{24} < 5^{12} < 10^8$

(D) $10^8 < 5^{12} < 2^{24}$ $10^8 < 5^{12} < 2^{24}$

(E) $10^8 < 2^{24} < 5^{12}$ $10^8 < 2^{24} < 5^{12}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Since all of the exponents are multiples of four, we can simplify the problem by taking the fourth root of each number. Evaluating we get $10^2=100$, $5^3=125$, and $2^6=64$. Since $64<100<125$, it follows that $\boxed{\textbf{(A)}\ 2^{24}<10^8<5^{12}}$ is the correct answer.

由于所有指数都是 4 的倍数，取四次根简化：$10^2=100$、$5^3=125$、$2^6=64$。因为 $64<100<125$，所以 $\boxed{\textbf{(A)}\ 2^{24}<10^8<5^{12}}$ 是正确答案。

Topics

AL.008 Exponents & radicals