2006/07 AHSMC Part 1 Problem 11

Problem 11 of 16

If $k$ is a rational number such that $\sqrt[3]{9\sqrt{3} - 11\sqrt{2}} = \sqrt{3} + k\sqrt{2}$ , then the value of $k$ is

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We have $9\sqrt{3} - 11\sqrt{2} = \bigl(\sqrt{3} + k\sqrt{2}\bigr)^3 = 3\sqrt{3} + 9k\sqrt{2} + 6k^2\sqrt{3} + 2k^3\sqrt{2}$ . From $9 = 3 + 6k^2$ , we have $k = \pm 1$ . From $-11 = 9k + 2k^3$ , $k < 0$ . Hence $k = -1$ . The answer is (b).

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