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Berndt, Bruce; Chan, Heng; Zhang, Liang-Cheng (1998). "Radicals and units in Ramanujan's work"
Theorem 2.1.
(a) $(\sqrt[3]{5}-\sqrt[3]{4})^{1 / 2}=\frac{1}{3}(\sqrt[3]{2}+\sqrt[3]{20}-\sqrt[3]{25})$
(b) $(\sqrt[3]{28}-\sqrt[3]{27})^{1 / 2}=\frac{1}{3}(\sqrt[3]{98}-\sqrt[3]{28}-1)$
(c) $\left(\sqrt[5]{\frac{1}{5}}+\sqrt[5]{\frac{4}{5}}\right)^{1 / 2}=(1+\sqrt[5]{2}+\sqrt[5]{8})^{1 / 5}$
(d) $\left(\sqrt[5]{\frac{1}{5}}+\sqrt[5]{\frac{4}{5}}\right)^{1 / 2}=\sqrt[5]{\frac{16}{125}}+\sqrt[5]{\frac{8}{125}}+\sqrt[5]{\frac{2}{125}}-\sqrt[5]{\frac{1}{125}}$,
(e) $\left(\sqrt[5]{\frac{32}{5}}-\sqrt[5]{\frac{27}{5}}\right)^{1 / 3}=\sqrt[5]{\frac{1}{25}}+\sqrt[5]{\frac{3}{25}}-\sqrt[5]{\frac{9}{25}}$
(f) $\left(\frac{3+2 \sqrt[4]{5}}{3-2 \sqrt[4]{5}}\right)^{1 / 4}=\frac{\sqrt[4]{5}+1}{\sqrt[4]{5}-1}$
(g) $(7 \sqrt[3]{20}-19)^{1 / 6}=\sqrt[3]{\frac{5}{3}}-\sqrt[3]{\frac{2}{3}}$
(h) $\left(4 \sqrt[3]{\frac{2}{3}}-5 \sqrt[3]{\frac{1}{3}}\right)^{1 / 8}=\sqrt[3]{\frac{4}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{1}{9}}$
(i) $(\sqrt[3]{2}-1)^{1 / 3}=\sqrt[3]{\frac{1}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}} .$
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