1990 AIME Problems/Problem 2

Problem

Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$ .

Solution

Suppose that $52+6\sqrt{43}$ is in the form of $(a + b\sqrt{43})^2$ . FOILing yields that $52 + 6\sqrt{43} = a^2 + 43b^2 + 2ab\sqrt{43}$ . This implies that $a$ and $b$ equal one of $\pm1, \pm3$ . The possible sets are $(3,1)$ and $(-3,-1)$ ; the latter can be discarded since the square root must be positive. This means that $52 + 6\sqrt{43} = (\sqrt{43} + 3)^2$ . Repeating this for $52-6\sqrt{43}$ , the only feasible possibility is $(\sqrt{43} - 3)^2$ .

Rewriting, we get $(\sqrt{43} + 3)^3 - (\sqrt{43} - 3)^3$ . Using the difference of cubes, we get that $[\sqrt{43} + 3\ - \sqrt{43} + 3]\ [(43 + 6\sqrt{43} + 9) + (43 - 9) + (43 - 6\sqrt{43} + 9)]$ $= (6)(3 \cdot 43 + 9) = \boxed{828}$ .

1990 AIME (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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1990 AIME Problems/Problem 2

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