Question

Complex Numbers Evaluate the expression and write in the form$a+bi$.

1. $\frac{4+2i}{2-i}$
   
2. $\left(1-\sqrt{-1}\right)\left(1+\sqrt{-1}\right)$
   

Short answer

1. The simplified form of the given expression is $1.2+1.6i$.
2. The simplified form of the given expression is width="39">2+0i.

Step-by-step solution

Part a. Step 1. Definition of complex number.

The general form of a complex number is $z=a+bi$, where width="134">i=−1⇒i2=−1.

Part a. Step 2. Given Information.

The given expression is:

$\frac{4+2i}{2-i}$

Part a.  Step 3. Simplify the given expression.

The given expression can be written as:

$$\begin{gathered} \frac{4+2i}{2-i}=\frac{4+2i}{2-i}\times \frac{2+i}{2+i} \\ =\frac{\left(4\right)\left(2\right)+\left(4\right)\left(i\right)+\left(2i\right)\left(2\right)+\left(2i\right)\left(i\right)}{{\left(2\right)}^2-{\left(i\right)}^2}\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \\ =\frac{8+4i+4i+2i^2}{4-i^2} \\ =\frac{8+8i+2\left(-1\right)}{4-\left(-1\right)}\left[\because i^2=-1\right] \end{gathered}$$

Further simplify the above expression.

$$\begin{gathered} \frac{4+2i}{2-i}=\frac{8+8i-2}{4+1} \\ =\frac{6+8i}{5} \\ =\frac{6}{5}+\frac{8i}{5} \\ =1.2+1.6i \end{gathered}$$

Thus, the simplified form of the given expression is $1.2+1.6i$.

Part b. Step 1. Definition of complex number.

The general form of a complex number is $z=a+bi$, where width="134">i=−1⇒i2=−1.

Part b. Step 2. Given Information.

The given expression is:

$\left(1-\sqrt{-1}\right)\left(1+\sqrt{-1}\right)$

Part b. Step 3. Simplify the given expression.

The given expression can be written as:

$$\begin{gathered} \left(1-\sqrt{-1}\right)\left(1+\sqrt{-1}\right)=\left(1-i\right)\left(1+i\right)\left[\because i=\sqrt{-1}\right] \\ =1^2-i^2\left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right] \\ =1-\left(-1\right)\left[\because i^2=-1\right] \\ =1+1 \\ =2 \end{gathered}$$

The complex form$a+bi$ of is $2+0i$.

Thus, the simplified form of the given expression is $2+0i$.