31 + $\sqrt[4]{21}$

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Solution

$\sqrt{31 + 4 \sqrt{21}} = \sqrt{31 + 4 \sqrt{7 \times 3}} = \sqrt{(28 + 3) + 2 \times 2 \sqrt{7} \times \sqrt{3}} = \sqrt{{(2 \sqrt{7})}^{2} + {(\sqrt{3})}^{2} + 2 \times 2 \sqrt{7} \times \sqrt{3}} = \sqrt{{(2 \sqrt{7} + \sqrt{3})}^{2}} = 2 \sqrt{7} + \sqrt{3}$

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