Which of the following is closest to √65-√63?

1987 AMC 12 Problem 19

Attempt Problem 19 of the 1987 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1987 AMC 12 solutions, or check the answer key.

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19.

Which of the following is closest to

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

0.12

0.13

0.14

0.15

0.16

Rationalize the difference of the two square roots

Compare

\sqrt{65}+\sqrt{63}

with

16

to decide which side of

0.125

the result lies on

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.