Solution to Problem 1

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Correct Answer:

\(\sqrt[3]{10+\sqrt{108}}-\sqrt[3]{-10+\sqrt{108}} = 2\)

The above answers were given by Nagendra P, Mnajunath M. R and Harshith Kumar N V.

However the correct justification were given by Nagendra P and Manjunath M. R. If you did not have calculator, you coud have proved that the value of the expression is 2 in two ways:

First way: This is due to Nagendra P and Manjunath M. R.

\(10+\sqrt{108} = (1+\sqrt{3})^3\) and \(-10+\sqrt{108} = (-1+\sqrt{3})^3\) Hence \(\sqrt[3]{10+\sqrt{108}}-\sqrt[3]{-10+\sqrt{108}} = (1+\sqrt{3})-(-1+\sqrt{3}) = 2\)

Second way: Setting \(x = \sqrt[3]{10+\sqrt{108}}-\sqrt[3]{-10+\sqrt{108}}\), cubing both sides and then applying \((a-b)^3\) formula, we come to the equation

\(x^3 +6x - 20 =0\)

which indicates that \(\sqrt[3]{10+\sqrt{108}}-\sqrt[3]{-10+\sqrt{108}}\) is a root of the above equation. It is also a mere inspection that \(x=2\) is also a root of the above equation and the above equation has no other real roots. Hence we must have

\(\sqrt[3]{10+\sqrt{108}}-\sqrt[3]{-10+\sqrt{108}}\) = 2.