What is $(26+15(3)^{1/2})^{1/3}+(26-15(3)^{1/2})^{1/3}$?
Solution 1:
$26 + 15 \sqrt{3} = 8 + 12 \sqrt{3} + 18 + 3 \sqrt{3} = 2^3 + 3 \cdot 2^2 \sqrt{3} + 3 \cdot 2 \sqrt{3}^2 + \sqrt{3}^3 = (2 + \sqrt{3})^3.$
$26 - 15 \sqrt{3} = 8 - 12 \sqrt{3} + 1 8- 3 \sqrt{3} = 2^3 - 3 \cdot 2^2 \sqrt{3} + 3 \cdot 2 \sqrt{3}^2 - \sqrt{3}^3 = (2 - \sqrt{3})^3.$
$(26 + 15 \sqrt{3})^{\frac{1}{3}} + (26 - 15 \sqrt{3})^{\frac{1}{3}} = 2 + \sqrt{3} + 2 - \sqrt{3} = 4.$