Question#01: Evaluate
\((i)\)
\[{{i}^{7}}\]
Solution:
\[\begin{align}
& ={{i}^{7}} \\
& ={{i}^{6}}\times i \\
& ={{\left( {{i}^{2}} \right)}^{3}}\times i \\
& ={{\left( -1 \right)}^{3}}\times i \\
& =-1\times i \\
& =-i \\
\end{align}\]
\((ii)\)
\[{{i}^{50}}\]
Solution:
\[\begin{align}
& ={{i}^{50}} \\
& ={{\left( {{i}^{2}} \right)}^{25}} \\
& ={{\left( -1 \right)}^{25}} \\
& =-1 \\
\end{align}\]
\((iii)\)
\[{{i}^{12}}\]
Solution:
\[\begin{align}
& ={{i}^{12}} \\
& ={{\left( {{i}^{2}} \right)}^{6}} \\
& ={{\left( -1 \right)}^{6}} \\
& =1 \\
\end{align}\]
\((iv)\)
\[{{\left( -i \right)}^{8}}\]
Solution:
\[\begin{align}
& ={{\left( -i \right)}^{8}} \\
& ={{i}^{8}} \\
& ={{\left( {{i}^{2}} \right)}^{4}} \\
& ={{\left( -1 \right)}^{4}} \\
& =1 \\
\end{align}\]
\((v)\)
\[{{\left( -i \right)}^{5}}\]
Solution:
\[\begin{align}
& ={{\left( -i \right)}^{5}} \\
& =-{{i}^{5}} \\
& =-{{i}^{4}}\times i \\
& =-{{\left( {{i}^{2}} \right)}^{2}}\times i \\
& =-{{\left( -1 \right)}^{2}}\times i \\
& =-\left( 1 \right)\times i \\
& =-i \\
\end{align}\]
\((vi)\)
\[{{i}^{27}}\]
Solution:
\[\begin{align}
& ={{i}^{27}} \\
& ={{i}^{26}}\times i \\
& ={{\left( {{i}^{2}} \right)}^{13}}\times i \\
& ={{\left( -1 \right)}^{13}}\times i \\
& =\left( -1 \right)\times i \\
& =-i \\
\end{align}\]
Question#2: Write the conjugate of the following numbers.
\[\begin{align}
\left( \text{i} \right)\qquad&2+3i \\
& \text{Conjugate}=2-3i \\
\left( \text{ii} \right)\qquad&3-5i \\
& \text{Conjugate}=3+5i \\
\left( \text{iii} \right)\qquad&-i \\
& \text{Conjugate}=i \\
\left( \text{iv} \right)\qquad&-3+4i \\
& \text{Conjugate}=-3-4i \\
\left( \text{v} \right)\qquad&-4-i \\
& \text{Conjugate}=-4+i \\
\left( \text{vi} \right)\qquad&i-3 \\
& \text{Conjugate}=-i-3 \\
\end{align}\]
Question#3: Write the real and imaginary parts of the following numbers.
\[\begin{align}
\left( \text{i} \right)\qquad &1+i \\
& \text{Real}=1 \\
& \text{Imaginary}=1 \\
\left( \text{ii} \right) \qquad&-1+2i \\
& \text{Real}=-1 \\
& \text{Imaginary}=2 \\
\left( \text{iii} \right) \qquad&-3i+2 \\
& \text{Real}=2 \\
& \text{Imaginary}=-3 \\
\left( \text{iv} \right)\qquad &-2-2i \\
& \text{Real}=-2 \\
& \text{Imaginary}=-2 \\
\left( \text{v} \right) \qquad&-3i \\
& \text{Real}=0 \\
& \text{Imaginary}=-3 \\
\left( \text{vi} \right) \qquad&2+0i \\
& \text{Real}=2 \\
& \text{Imaginary}=0 \\
\end{align}\]
Question#4: Find the value of \(x\) and \(y\) if \(x+iy+1=4-3i\).
Solution:
\[\begin{align}
& x+iy+1=4-3i \\
& x+iy=4-3i-1 \\
& x+iy=3-3i \\
& \text{Comparing real and imaginary parts} \\
& x=3,\quad y=-3 \\
\end{align}\]
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