1980 AHSME Problems/Problem 27

Problem

The sum $\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ equals

$\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad \text{(D)}\ \sqrt[3]{2}\qquad \text{(E)}\ \text{none of these}$

Solution

Let $y = \sqrt[3] {5+2\sqrt{13}}$ , $z = \sqrt[3]{5-2\sqrt{13}}$ , and $x = \sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}} = y + z$ .

Then $x^3 = (y+z)^3 = y^3 + 3y^2z + 3yz^2 + z^3 = y^3 + z^3 + 3yz(y+z) = y^3 + z^3 + 3yzx$ .

Note that $y^3 + z^3 = 5 + 2\sqrt{13} + 5 - 2\sqrt{13} = 10$ and $yz = \sqrt[3]{\left(5 + 2\sqrt{13}\right)\left(5-2\sqrt{13}\right)} = \sqrt[3]{5^2 -\left(2\sqrt{13}\right)^2} = \sqrt[3]{-27} = -3$ .

So $x^3 = 10 + 3 \cdot -3 \cdot x = 10-9x$ , that is $x^3+9x-10=0$ . This factors to $(x-1)(x^2+x+10) = 0$ , which has $x=1$ as its only real solution.

Therefore, $x = 1$ and the answer is $\boxed{(\textbf{E})\ \textrm{none of these}}$ .

1980 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 26	Followed by Problem 28
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1980 AHSME Problems/Problem 27

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