Need help on this math question about magnitudes

Just use the formula for |z|^2.

Clean up your question. Put your LaTex in a LaTex box  (found on the ribbon)

$z\overline{w} = 6 + 8i\\ \overline{z\overline{w}} = \overline{6+8i}\\ \overline{z}w = 6 - 8i$

And then, we know that for any complex number z and w, $z\overline{z} = |z|^2$ and $|z||w| = |zw|$.

$\quad|zw|^2 \\= |z|^2 |w|^2 \\= z\overline{z}w\overline{w} \\= (z\overline{w})(w\overline z) \\= (6 + 8i)(6 - 8i) \\= 10$

$\quad|z + w|^2 \\= (z + w)\left(\overline{z + w}\right) \\= (z + w)\left(\overline z + \overline w\right)\\ = z\overline z + w\overline w + z\overline w + w\overline z \\= |z|^2 + |w|^2 + z\overline w + \overline zw \\= 5^2 + 2^2 + (6 + 8i) + (6- 8i) \\= 41$

I will leave the remaining ones to you.