====== Question 8, Exercise 1.1 ======
Solutions of Question 8 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.
=====Question 8(i)=====
Express the $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$ in the standard form of $a+ib.$ 
====Solution====
\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\
&=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\
&=\dfrac{\left( 3+4+2i-6i \right)+\left( 8+1+4i-2i \right)}{\left( 6-2+3i+4i \right)}\\
&=\dfrac{\left( 7-4i \right)+\left( 9+2i \right)}{4+7i}\\
&=\dfrac{16-2i}{4+7i}\\
&=\dfrac{16-2i}{4+7i}\times \dfrac{4-7i}{4-7i}\\
&=\dfrac{\left( 64-14 \right)-\left( 112+8 \right)i}{16+49}\\
&=\dfrac{50-120i}{65}\\
&=\dfrac{10-24i}{13}\\
&=\dfrac{10}{13}-\dfrac{24i}{13}\end{align}

=====Question 8(ii)=====
Express the $\dfrac{2+\sqrt{-9}}{-5-\sqrt{-16}}$ in the standard form of $a+ib.$ 
====Solution====
\begin{align}\dfrac{2+\sqrt{-9}}{-5-\sqrt{-16}}&=\dfrac{2+3i}{-5-4i}\\
&=\dfrac{2+3i}{-5-4i}\times \dfrac{-5+4i}{-5+4i}\\
&=\dfrac{\left( -10-12 \right)+\left( 8-15 \right)i}{25+16}\\
&=\dfrac{-22-7i}{41}\\
&=\dfrac{-22}{41}-\dfrac{7i}{41}\end{align}


=====Question 8(iii)=====
Express the $\dfrac{{{\left( 1+i \right)}^{2}}}{4+3i}$ in the standard form of $a+ib.$ 
====Solution====
\begin{align}\dfrac{\left( 1+i \right)\left( 1+i \right)}{4+3i}&=\dfrac{1-1+i+i}{4+3i}\\
&=\dfrac{2i}{4+3i}\\
&=\dfrac{2i}{4+3i}\times \dfrac{4-3i}{4-3i}\\
&=\dfrac{6+8i}{16+9}\\
&=\dfrac{6+8i}{25}\\
&=\dfrac{6}{25}+\dfrac{8i}{25}\end{align}

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