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Algebra Level 3

If the following expression has a positive integer positive value, what is it?

25 8 + 11 2 4 3 + 25 8 − 11 2 4 3 \large \sqrt[3]{\frac{25}{8} + \frac{11\sqrt2}{4}} + \sqrt[3]{\frac{25}{8} - \frac{11\sqrt2}{4}}


The answer is 1.

Relue Tamref by Relue Tamref , 21, Brazil

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1 solution

Sándor Daróczi
Sep 4, 2017

Observe that

( 1 2 + 2 ) 3 = 1 8 + 3 ⋅ 1 4 ⋅ 2 + 3 ⋅ 1 2 ⋅ 2 + 2 2 = 25 8 + 11 2 4 (\frac{1}{2}+\sqrt{2})^3 = \frac{1}{8} + 3 \cdot \frac{1}{4} \cdot \sqrt{2} + 3 \cdot \frac{1}{2} \cdot 2 + 2\sqrt{2} = \frac{25}{8}+\frac{11\sqrt{2}}{4}

and similarly

( 1 2 − 2 ) 3 = 1 8 − 3 ⋅ 1 4 ⋅ 2 + 3 ⋅ 1 2 ⋅ 2 − 2 2 = 25 8 − 11 2 4 (\frac{1}{2}-\sqrt{2})^3 = \frac{1}{8} - 3 \cdot \frac{1}{4} \cdot \sqrt{2} + 3 \cdot \frac{1}{2} \cdot 2 - 2\sqrt{2} = \frac{25}{8}-\frac{11\sqrt{2}}{4}

Hence the given expression equals to

1 2 + 2 + 1 2 − 2 = 1 \frac{1}{2}+\sqrt{2}+\frac{1}{2}-\sqrt{2}=1 .

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