Let $z=-8+15i$ and $w=6-8i.$ Compute \[\dfrac{z\overline z}{w\overline w},\]where the bar represents the complex conjugate.
+128408 \(\dfrac{z\overline z}{w\overline w}\)
So we have
[-8 + 15 i ] [ -8 - 15i ] 64 - 225i^2 64 - 225(-1) 289
_________________ = ___________ = ___________ = ______
[ 6 - 8i ] [ 6 + 8i ] 36 - 64i^2 36 - 64(-1) 100
+9466 This video really helped me understand: https://www.youtube.com/watch?v=BZxZ_eEuJBM
If \(z=-8+15i\) then \(\overline{z}=-8-15i\)
If \(w=6-8i\) then \(\overline{w}=6+8i\)
And so...
\(\dfrac{z\overline{z}}{w\overline{w}}\ =\ \dfrac{(-8+15i)(-8-15i)}{(6-8i)(6+8i)}\ =\ \dfrac{(-8)^2-(15i)^2}{(6)^2-(8i)^2}\ =\ \dfrac{64+225}{36+64}\ =\ \dfrac{289}{100}\)
Just like CPhill found. 
+26367 Let
\(z=-8+15i \text{ and }w=6-8i.\)
Compute
\(\dfrac{z\overline z}{w\overline w}\),
where the bar represents the complex conjugate.
\(\begin{array}{|rcl|rcl|} \hline z\overline z &=& |z|^2 & w\overline w &=& |w|^2 \\ \hline z &=&-8+15i & w&=&6-8i \\ |z| &=& \sqrt{8^2+15^2} & |w| &=& \sqrt{6^2+8^2} \\ |z|^2 &=& 8^2+15^2 & |w|^2 &=& 6^2+8^2 \\ \hline \mathbf{\dfrac{z\overline z}{w\overline w}} &=& \dfrac{|z|^2}{|w|^2} \\ &=& \dfrac{8^2+15^2}{6^2+8^2 } \\ &=& \mathbf{ \dfrac{289}{100} } \\ \hline \end{array} \)
