1987 AMC 12 第 19 题

Question

下列哪个数最接近

\sqrt{65}-\sqrt{63}

？

Which of the following is closest to

\sqrt{65}-\sqrt{63}?

0.12

0.13

0.14

0.15

0.16

将两个平方根之差有理化

Rationalize the difference of the two square roots

将

\sqrt{65}+\sqrt{63}

与

16

比较，以判断结果位于

0.125

的哪一侧

Compare

\sqrt{65}+\sqrt{63}

with

16

to decide which side of

0.125

the result lies on

有理化得

\sqrt{65}-\sqrt{63}=\frac2{\sqrt{65}+\sqrt{63}}\text{。}

分母小于

16

，因为它的平方为

128+2\sqrt{4095}\lt256

。所以该值大于

\frac{2}{16}=0.125

。此外，两个根式都大于

7.5

，所以该值小于

\frac{2}{15}<0.134

。因此，它与

0.13

的距离小于与其他任何选项的距离。

所以正确答案是 B。

Rationalizing gives

\sqrt{65}-\sqrt{63}=\frac2{\sqrt{65}+\sqrt{63}}.

The denominator is less than

16,

because its square is

128+2\sqrt{4095}\lt256.

Thus the value is greater than

\frac{2}{16}=0.125.

Also each radical exceeds

7.5,

so the value is less than

\frac{2}{15}<0.134.

It is therefore closer to

0.13

than to any other choice.

Thus the correct answer is B.

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