<s>[INST] Operations with surds (Pre-calculus) I have tried multiple ways to solve this:
$$ \left (2\sqrt 8 - 3\sqrt 5 \right)^2 $$
(original screenshot)
and my answer was way off. Can someone show me their method? Thanks.
 [/INST] The simplest way to prevent any careless mistakes is to apply the identity
$$(a - b)^2 = a^2 + b^2 - 2ab$$
Substitute $a = 2\sqrt{8}$ and $b = 3\sqrt{5}$. Then we have 
$$a^2 = (2\sqrt8)^2 = 4\cdot8 = 32$$
and
$$b^2 = (3\sqrt{5})^2 = 9\cdot5 = 45$$
and
$$2ab = 2\cdot2\sqrt8\cdot3\sqrt5=12\sqrt{40}$$
So we have
$$(2\sqrt{8} - 3\sqrt{5})^2 = 32 + 45 - 12\sqrt{40} = 77 - 12\sqrt{40}$$
</s>