2014 Singapore MO (Junior Rd 1) Problem 4

2014 SMO (Junior-Rd1) 4. Find the value of \displaystyle\frac{1}{1-\sqrt[4]{5}} + \frac{1}{1+\sqrt[4]{5}} + \frac{2}{1+\sqrt{5}}.

(A) -1   (B) 1   (C) -\sqrt{5}   (D) \sqrt{5}   (E) None of the above

To simplify fractions, we can put them all under the same denominator and hope the expressions in the numerator and denominator simplify nicely. This straightforward method makes the expansion and simplification quite messy (even though it can be solved from here): \text{Expression} = \displaystyle\frac{(1+\sqrt[4]{5})(1+\sqrt{5}) + (1-\sqrt[4]{5})(1+\sqrt{5}) + 2(1-\sqrt[4]{5})(1+\sqrt[4]{5})}{(1-\sqrt[4]{5})(1+\sqrt[4]{5})(1+\sqrt{5})}. There is no need to put them all under the same denominator at the same time: we can combine 2 fractions at each step— there could be some simplification after each step. Indeed, if we notice that the denominators 1 - \sqrt[4]{5} and 1+\sqrt[4]{5} are conjugates and hence simplify nicely when multiplied together,

\begin{aligned} \frac{1}{1-\sqrt[4]{5}} + \frac{1}{1+\sqrt[4]{5}} &= \frac{1+\sqrt[4]{5} + 1-\sqrt[4]{5}}{(1-\sqrt[4]{5})(1+\sqrt[4]{5})} \\ &= \frac{2}{1-\sqrt{5}}, \end{aligned}

Hence the expression in the question simplifies to

\begin{aligned} \frac{2}{1-\sqrt{5}} + \frac{2}{1+\sqrt{5}} &= \frac{2(1+\sqrt{5}) + 2(1-\sqrt{5})}{(1-\sqrt{5})(1+\sqrt{5})} \\ &= \frac{4}{1-5} = -1. \end{aligned}

The answer is (A).

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