Question

z1=1+√3i z2=2-5i find (z1 z2)-1​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{-\:368\:+\:500\:\sqrt{3}}{8656}\:+\:\dfrac{(\:380\:-\:152\:\sqrt{3}\:)\:i}{8656}}}}$

Step-by-step-explanation:

We have given two complex numbers.

$\displaystyle{\sf\:z_1\:=\:1\:+\:\sqrt{3}\:i}$

$\displaystyle{\sf\:z_2\:=\:2\:-\:5\:i}$

We have to find the value of

$\displaystyle{\sf\:(\:z_1\:z_2\:)^{-\:1}}$

Now,

$\displaystyle{\sf\:z_1\:z_2\:=\:(\:1\:+\:\sqrt{3}\:i\:)\:(\:2\:-\:5\:i\:)}$

$\displaystyle{\implies\sf\:z_1\:z_2\:=\:1\:(\:2\:-\:5i\:)\:+\:\sqrt{3}i\:(\:2\:-\:5i\:)}$

$\displaystyle{\implies\sf\:z_1\:z_2\:=\:2\:-\:5i\:+\:2\:\sqrt{3}i\:-\:5\:\sqrt{3}\:i^2}$

$\displaystyle{\implies\sf\:z_1\:z_2\:=\:2\:+\:2\:\sqrt{3}i\:-\:5i\:-\:5\:\sqrt{3}\:(-\:1)}$

$\displaystyle{\implies\sf\:z_1\:z_2\:=\:2\:+\:2\:\sqrt{3}i\:-\:5i\:+\:5\:\sqrt{3}}$

$\displaystyle{\implies\sf\:z_1\:z_2\:=\:2\:+\:+\:5\:\sqrt{3}\:-\:5i\:+\:2\:\sqrt{3}i}$

$\displaystyle{\implies\:\boxed{\blue{\sf\:z_1\:z_2\:=\:2\:+\:5\:\sqrt{3}\:-\:(\:5\:-\:2\:\sqrt{3}\:)\:i}}}$

Now, we know that,

$\displaystyle{\boxed{\pink{\sf\:a^{-\:1}\:=\:\dfrac{1}{a}\:}}}$

$\displaystyle{\therefore\:\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{1}{z_1\:z_2}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{1}{2\:+\:5\:\sqrt{3}\:-\:(\:5\:-\:2\:\sqrt{3}\:)i}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{1}{2\:+\:5\:\sqrt{3}\:-\:(\:5\:-\:2\:\sqrt{3}\:)i}\:\times\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{(\:2\:+\:5\:\sqrt{3}\:-\:(\:5\:-\:2\:\sqrt{3}\:)i\:)\:(\:2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)\:i\:)}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{(\:2\:+\:5\:\sqrt{3}\:)^2\:-\:[5\:-\:2\:\sqrt{3}\:\:)\:i]^2}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{2^2\:+\:2\:\times\:2\:\times\:5\:\sqrt{3}\:+\:(\:5\:\sqrt{3}\:)^2\:-\:[5^2\:-\:2\:\times\:5\:\times\:2\:\sqrt{3}\:+\:(\:2\:\sqrt{3}\:)^2]\:i^2}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{4\:+\:20\:\sqrt{3}\:+\:25\:\times\:3\:-\:(\:25\:-\:20\:\sqrt{3}\:+\:4\:\times\:3\:)\:-\:1}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{4\:+\:20\:\sqrt{3}\:+\:75\:+\:(\:25\:-\:20\:\sqrt{3}\:+\:12\:)}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{4\:+\:20\:\sqrt{3}\:+\:75\:+\:25\:-\:20\:\sqrt{3}\:+\:12}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{20\:\sqrt{3}\:+\:20\:\sqrt{3}\:+\:100\:+\:4\:+\:12}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{40\:\sqrt{3}\:+\:100\:+\:16}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:(\:5\:-\:2\:\sqrt{3}\:)i}{116\:+\:40\:\sqrt{3}}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:5i\:-\:2\:\sqrt{3}i}{116\:+\:40\:\sqrt{3}}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{2\:+\:5\:\sqrt{3}\:+\:5i\:-\:2\:\sqrt{3}i}{116\:+\:40\:\sqrt{3}}\:\times\:\dfrac{116\:-\:40\:\sqrt{3}}{116\:-\:40\:\sqrt{3}}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{(\:2\:+\:5\:\sqrt{3}\:+\:5i\:-\:2\:\sqrt{3}i\:)\:(\:116\:-\:40\:\sqrt{3}\:)}{(\:116\:+\:40\:\sqrt{3}\:)\:(\:116\:-\:40\:\sqrt{3}\:)}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{116\:(\:2\:+\:5\:\sqrt{3}\:+\:5i\:-\:2\:\sqrt{3}i\:)\:-\:40\:\sqrt{3}\:(\:2\:+\:5\:\sqrt{3}\:+\:5i\:-\:2\:\sqrt{3}i\:)}{(\:116\:)^2\:-\:(\:40\:\sqrt{3}\:)^2}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{232\:+\:580\:\sqrt{3}\:+\:580i\:-\:232\:\sqrt{3}i\:-\:80\:\sqrt{3}\:-\:200\:\times\:3\:-\:200i\:+\:80\:\sqrt{3}i}{13456\:-\:1600\:\times\:3}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{232\:-\:600\:+\:580\:\sqrt{3}\:-\:80\:\sqrt{3}\:+\:580i\:-\:200i\:-\:232\:\sqrt{3}\:i\:+\:80\:\sqrt{3}i}{13456\:-\:4800}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{-\:368\:+\:500\:\sqrt{3}\:+\:380i\:-\:152\:\sqrt{3}i}{8656}}$

$\displaystyle{\implies\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{-\:368\:+\:500\:\sqrt{3}\:+\:(\:380\:-\:152\:\sqrt{3}\:)i}{8656}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:(\:z_1\:z_2\:)^{-\:1}\:=\:\dfrac{-\:368\:+\:500\:\sqrt{3}}{8656}\:+\:\dfrac{(\:380\:-\:152\:\sqrt{3}\:)\:i}{8656}}}}}$