If $$a^{4} + \frac{1}{a^{4}} = 50$$, then find the value of $$a^{3} + \frac{1}{a^{3}}$$

$$a^{4} + \frac{1}{a^{4}} = 50$$

$$a^{4} + \frac{1}{a^{4}} + 2= 50 + 2$$

$$(a^2+\frac{1}{a^2})^2=52$$

$$(a^2+\frac{1}{a^2})=\sqrt{52}$$

$$a^2+\frac{1}{a^2} + 2 = \sqrt{52} + 2$$

$$(a + \frac{1}{a})^2 = \sqrt{52} + 2$$

$$(a + \frac{1}{a}) = \sqrt{\sqrt{52} + 2}$$

$$a^{3} + \frac{1}{a^{3}} = (a + b)^3 + 3ab(a + b)$$

=$$(\sqrt{\sqrt{52} + 2})^3 + \sqrt{\sqrt{52} + 2}$$

=$$(\sqrt{2\sqrt{13} + 2})^3 + \sqrt{2\sqrt{13} + 2}$$

=$$\sqrt{2\sqrt{13} + 2}(1 + (\sqrt{2\sqrt{13} + 2})^2)$$

=$$\sqrt{2\sqrt{13} + 2}(1 + {2\sqrt{13} + 2})$$

=$$\sqrt{2(\sqrt{13} + 1})(3 + {2\sqrt{13}})$$