\(\displaystyle In\ order\ to\ simplify\ (4-7i)(1-2i)\ we\ must\ multiply\ the\ binomials\ using\ the\ FOIL\ method.\)

\(\displaystyle F\ is\ for\ first\ times\ first: =4*1=4\)

\(\displaystyle O\ is\ for\ outside\ times\ outside =4*(-2i)=-8i\)

\(\displaystyle I\ is\ for\ inside\ times\ inside=(-7i)*1=-7i\)

\(\displaystyle L\ is\ for\ last\ times\ last =(-7i)*(-2i)=14i^2\)

\(\displaystyle Now\ we\ combine\ like\ terms\ and\ simplify:\)

\(\displaystyle 4-8i-7i+14i^2\)

\(\displaystyle ^* Remember\ that\ i^2=-1\)

\(\displaystyle 4-15i+14(-1)=-10-15i\)

\(\displaystyle When\ a\ binomial\ is\ squared\ we\ must\ write\ it\ out\ twice\ and\ perform\ the\ FOIL\ method.\)

\(\displaystyle (5-8i)^2=(5-8i)(5-8i)\)

\(\displaystyle F\ is\ for\ first\ times\ first = 5*5=25\)

\(\displaystyle O\ is\ for\ outside\ times\ outside = 5*8i=40i\)

\(\displaystyle I\ is\ for\ inside\ times\ inside: 8i*5=40i\)

\(\displaystyle L\ is\ for\ last\ times\ last = 8i*8i=64i^2=64(-1)=-64\)

Now we put each of these together and combine like terms: 

\(\displaystyle 25-40i-40i-64=-39-80i\)

\(\displaystyle \frac{\sqrt{-16}}{\sqrt{-8}}\)

Take i (the square root of -1) out of both radicals then divide.

\(\displaystyle \frac{i\sqrt{16}}{i\sqrt{8}}\)

\(\displaystyle \frac{\sqrt{16}}{\sqrt{8}}\)

\(\displaystyle \sqrt{2}\)

\(\displaystyle \sqrt{-16}\cdot \sqrt{-25}\)

Take out i (the square root of -1) from both radicals and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

\(\displaystyle 4i\cdot 5i\)

\(\displaystyle 20i^{2}\)

Make i squared -1

\(\displaystyle 20(-1)\)

\(\displaystyle -20\)

\(\displaystyle \sqrt{-7}\cdot \sqrt{-4}\)

First, take out i (the square root of -1) from both radicals and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

\(\displaystyle i\sqrt{7}\cdot 2i\)

\(\displaystyle 2i^{2}\sqrt{7}\)

Change i squared to -1

\(\displaystyle 2(-1)\sqrt{7}\)

\(\displaystyle -2\sqrt{7}\)

\(\displaystyle \frac{\sqrt{-51}}{\sqrt{3}}\)

Take i (the square root of -1) out of the radical.

\(\displaystyle \frac{i\sqrt{51}}{\sqrt{3}}\)

\(\displaystyle i\sqrt{17}\)

\(\displaystyle \sqrt{5}\cdot \sqrt{-42}\)

Take out i (the square root of -1) from the radical and then multiply.

\(\displaystyle \sqrt{5}\cdot i\sqrt{42}\)

\(\displaystyle i\sqrt{210}\)

\(\displaystyle \sqrt{-45}\cdot \sqrt{72}\)

Take out i (the square root of -1) and then simplify before multiplying.

\(\displaystyle i\sqrt{45}\cdot \sqrt{72}\)

\(\displaystyle i\sqrt{9\cdot 5}\cdot \sqrt{36\cdot 2}\)

\(\displaystyle 3i\sqrt{5}\cdot 6\sqrt{2}\)

\(\displaystyle 18i\sqrt{10}\)

\(\displaystyle -\sqrt{-16}\cdot \sqrt{-40}\)

Take out i (the square root of -1) from both radicals and then multiply. You are not allowed to first multiply the radicals and then simplify because the roots are negative.

\(\displaystyle -i\sqrt{16}\cdot i\sqrt{40}\)

\(\displaystyle -4i\cdot i\sqrt{4\cdot 10}\)

\(\displaystyle -4i\cdot 2i\sqrt{10}\)

\(\displaystyle -8i^{2}\sqrt{10}\)

\(\displaystyle -8(-1)\sqrt{10}\)

\(\displaystyle 8\sqrt{10}\)

\(\displaystyle -\sqrt{-20}\cdot -\sqrt{32}\)

Take out i (the square root of -1) from the radical, simplify, and then multiply.

\(\displaystyle -i\sqrt{20}\cdot -\sqrt{32}\)

\(\displaystyle -i\sqrt{5\cdot 4}\cdot -\sqrt{16\cdot 2}\)

\(\displaystyle -2i\sqrt{5}\cdot -4\sqrt{2}\)

\(\displaystyle 8i\sqrt{10}\)