What is the value of $$\left(\sqrt{7+\sqrt{29}}-\sqrt{7-\sqrt{29}}\right)^2?$$ $$A.-26$$ $$B.\ 2\sqrt{29}$$ $$C.\ 14-4\sqrt{5}$$ $$D.\ 14$$ $$E.\ 14+4\sqrt{5}$$

The OA is C .

Experts, may you help me here and tell me how to find the correct answer? Please.

Hi M7MBA,
Let's take a look at your questions.

$$\left(\sqrt{7+\sqrt{29}}-\sqrt{7-\sqrt{29}}\right)^2 ... (i)$$
To evaluate it we will be using the formula:
$$\left(a-b\right)^2=a^2-2ab+b^2$$
Using this formula (i) can be written as:
$$=\left(\sqrt{7+\sqrt{29}}\right)^2-2\left(\sqrt{7+\sqrt{29}}\right)\left(\sqrt{7-\sqrt{29}}\right)+\left(\sqrt{7-\sqrt{29}}\right)^2$$
$$=\left(7+\sqrt{29}\right)-2\left(\sqrt{\left(7+\sqrt{29}\right)\left(7-\sqrt{29}\right)}\right)+\left(7-\sqrt{29}\right)$$ $$=\left(7+\sqrt{29}\right)-2\left(\sqrt{\left(7\right)^2-\left(\sqrt{29}\right)^2}\right)+\left(7-\sqrt{29}\right)$$
$$=\left(7+\sqrt{29}\right)-2\left(\sqrt{49-29}\right)+\left(7-\sqrt{29}\right)$$
$$=\left(7+\sqrt{29}\right)-2\left(\sqrt{20}\right)+\left(7-\sqrt{29}\right)$$
Combining like terms:
$$=7+7-\sqrt{29}+\sqrt{29}-2\left(\sqrt{20}\right)$$
$$=7+7-2\left(\sqrt{20}\right)$$
$$=14-2\left(\sqrt{20}\right)$$
$$=14-2\left(\sqrt{4\times5}\right)$$
$$=14-2\times2\sqrt{5}$$
$$=14-4\sqrt{5}$$

Therefore, Option C is correct.

Hope it helps.
I am available if you'd like any follow up.