Too much roots!

Algebra Level 4

The rationalization of the expression

1 5 + 2 + 1 \dfrac{1}{\sqrt{5} + \sqrt{2} + 1}

can be expressed as

− A 10 − B 5 + C 2 + D E , \dfrac{- A\sqrt{10} - B\sqrt{5} + C\sqrt{2} + D}{E},

where A A , B B , C C , D D , and E E are all positive integers .

Find the minimum value of A + B + C + D + E A + B + C + D + E .

This problem is part of the set " Xenophobia ."


The answer is 9.

Rindell Mabunga by Rindell Mabunga , 22, Philippines

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1 solution

Rindell Mabunga
Jun 8, 2016

1 5 + 2 + 1 \frac{1}{\sqrt{5} + \sqrt{2} + 1}

= 1 ( 5 + 2 ) + 1 × ( 5 + 2 ) − 1 ( 5 + 2 ) − 1 = \frac{1}{(\sqrt{5} + \sqrt{2}) + 1} \times \frac{(\sqrt{5} + \sqrt{2}) - 1}{(\sqrt{5} + \sqrt{2}) - 1}

= 5 + 2 − 1 6 + 2 10 = \frac{\sqrt{5} + \sqrt{2} - 1}{6 + 2\sqrt{10}}

= 5 + 2 − 1 2 ( 3 + 10 ) = \frac{\sqrt{5} + \sqrt{2} - 1}{2(3 + \sqrt{10})}

= 5 + 2 − 1 2 ( 3 + 10 ) × 3 − 10 3 − 10 = \frac{\sqrt{5} + \sqrt{2} - 1}{2(3 + \sqrt{10})} \times \frac{3 - \sqrt{10}}{3 - \sqrt{10}}

= 5 − 2 2 − 3 + 10 − 2 = \frac{\sqrt{5} - 2\sqrt{2} - 3 + \sqrt{10}}{-2}

= − 10 − 5 + 2 2 + 3 2 = \frac{ - \sqrt{10} - \sqrt{5} + 2\sqrt{2} + 3}{2}

A = 1 , B = 1 , C = 2 , D = 3 , E = 2 A = 1, B = 1, C = 2, D = 3, E = 2

A + B + C + D + E = 1 + 1 + 2 + 3 + 2 = 9 A + B + C + D + E = 1 + 1 + 2 + 3 + 2 = \boxed{9}

17 Helpful 2 Interesting 3 Brilliant 2 Confused

In the question it was given that gcd(A,E) =gcd(C,E) but it doesn't follows?

Sahasra Ranjan - 5 years ago

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Thanks. I've made the relevant edits to the problem.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the “line line line” menu in the top right corner. This will notify the problem creator who can fix the issues.

Brilliant Mathematics Staff - 5 years ago

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Oh sorry. Thanks for pointing out the error.

Rindell Mabunga - 5 years ago

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