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UKMT Senior Kangaroo, 2019

Algebra Difficulty 2.0 Junior Prove it United Kingdom

What is the value of $A^2 + B^3 + C^5$ , given that: $A = \sqrt[3]{16\sqrt{2}}$ , $B = \sqrt{9\sqrt[3]{9}}$ , $C = \left[\left(\sqrt[5]{2}\right)^{2}\right]^{2}$

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