The value of $$(52+6\sqrt{43})^{3/2} - (52-6\sqrt{43})^{3/2}$$ is

We are given the expression: $$(52+6\sqrt{43})^{3/2} - (52-6\sqrt{43})^{3/2}$$

Now, the term $$52+6\sqrt{43}$$ can be rewritten as:

$$52+6\sqrt{43}=9+43+2*3*\sqrt{43}=(3+\sqrt{43})^2$$

Similarly, we can rewrite the term $$52-6\sqrt{43}$$ as:

$$52-6\sqrt{43}=9+43-2*3*\sqrt{43}=(3-\sqrt{43})^2$$ or $$(\sqrt{43}-3)^2$$

But whenever we have $$\sqrt{52-6\sqrt{43}}$$, the correct answer would be $$(\sqrt{43}-3)$$ as the square root is defined in mathematics to give only a positive answer due to functional constraints put on it.

Now, we had:  $$(52+6\sqrt{43})^{3/2} - (52-6\sqrt{43})^{3/2}$$

$$=(\sqrt{52+6\sqrt{43}})^3 - (\sqrt{52-6\sqrt{43}})^3$$

$$=(3+\sqrt{43})^3-(\sqrt{43}-3)^3$$

$$=(27+\sqrt{43}^3+27\sqrt{43}+9*43)-(-27+\sqrt{43}^3+27\sqrt{43}-9*43)$$

$$=2(27+9*43)$$

$$=828$$

Hence, Option D is correct.