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$$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}} \right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$ $$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."

$$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}} \right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$ $$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."

$$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}} \right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$ $$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: https://faculty.math.illinois.edu/~berndt/jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."

$$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}} \right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$ $$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."

$$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}} \right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$ $$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: http://www.math.uiuc.edu/~berndt/jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."

$$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}} \right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$ $$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ \sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: https://faculty.math.illinois.edu/~berndt/jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."