1977 AHSME Problems/Problem 7

Problem 7

If $t = \frac{1}{1 - \sqrt[4]{2}}$, then $t$ equals

$\text{(A)}\ (1-\sqrt[4]{2})(2-\sqrt{2})\qquad \text{(B)}\ (1-\sqrt[4]{2})(1+\sqrt{2})\qquad \text{(C)}\ (1+\sqrt[4]{2})(1-\sqrt{2}) \qquad \\ \text{(D)}\ (1+\sqrt[4]{2})(1+\sqrt{2})\qquad \text{(E)}-(1+\sqrt[4]{2})(1+\sqrt{2})$


Solution

Solution by e_power_pi_times_i

$t = \dfrac{1}{1-\sqrt[4]{2}} = (\dfrac{1}{1-\sqrt[4]{2}})(\dfrac{1+\sqrt[4]{2}}{1+\sqrt[4]{2}}) = (\dfrac{1+\sqrt[4]{2}}{1-\sqrt{2}})(\dfrac{1+\sqrt{2}}{1+\sqrt{2}}) = \dfrac{(1+\sqrt[4]{2})(1+\sqrt{2})}{-1} = \text{(E)}-(1+\sqrt[4]{2})(1+\sqrt{2})$.

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