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Algebra Difficulty 3.0 AMC 10/12 Find the answer United Kingdom

The numbers $x$ , $y$ and $z$ are given by $x = \sqrt{12-3\sqrt{7}} - \sqrt{12+3\sqrt{7}}$ , $y = \sqrt{7-4\sqrt{3}} - \sqrt{7+4\sqrt{3}}$ and $z = \sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}}$ . What is the value of $xyz$ ?

(A) $1$ (B) $-6$ (C) $-8$ (D) $18$ (E) $12$

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.