Compute the absolute values of given complex numbers

For each of the following complex numbers, compute the absolute value.

Using the formula for the absolute value of a complex number we compute,
$|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}.$
Using the formula for the absolute value of a complex number we compute,
$|3+4i| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5.$
Using the fact that $\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}$ and the formula for the absolute value of a complex number we have,
$\left| \frac{1+i}{1-i} \right| = \frac{|1+i|}{|1-i|} = \frac{\sqrt{2}}{\sqrt{2}} = 1.$
Using $i^2 = -1$ and the formula for the absolute value of a complex number we have
$|1 + i + i^2| = |i| = \sqrt{1^2} = 1.$
Using that $i^4 = 1$ we have
$|i^7 + i^{10}| = |i^3 + i^2| = |-1 - i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}.$
We compute,
$|2(1-i) + 3(2+i)| = |2 - 2i + 6 + 3i| = |8 + i| = \sqrt{8^2 + 1^2} = \sqrt{65}.$