AMC10 2007 A

AMC10 2007 A · Q9

AMC10 2007 A · Q9. It mainly tests Systems of equations, Exponents & radicals.

Real numbers $a$ and $b$ satisfy the equations $3^a = 81^{b+2}$ and $125^b = 5^{a-3}$. What is $ab$?

实数$a$和$b$满足方程$3^a = 81^{b+2}$和$125^b = 5^{a-3}$。$ab$等于多少？

(A) -60 -60

(B) -17 -17

(D) 12 12

(E) 60 60

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): The given equations are equivalent, respectively, to $3^a = 3^{4(b+2)}$ and $5^{3b} = 5^{a-3}$. Therefore $a = 4(b+2)$ and $3b = a-3$. The solution of this system is $a=-12$ and $b=-5$, so $ab=60$.

答案（E）：给定的方程分别等价于 $3^a = 3^{4(b+2)}$ 和 $5^{3b} = 5^{a-3}$。因此 $a = 4(b+2)$ 且 $3b = a-3$。该方程组的解为 $a=-12$、$b=-5$，所以 $ab=60$。

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