Question

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

1. $\frac{8+3i}{4+3i}$
   
2. $\sqrt{-10}\cdot \sqrt{-40}$
   

Short answer

1. The simplified form of the given expression is $1.64-0.48i$.
2. The simplified form of the given expression is $-20+0i$.

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:

$\frac{8+3i}{4+3i}$

Part a. Step 3. Simplify the given expression.

The given expression can be written as:

$$\begin{gathered} \frac{8+3i}{4+3i}=\frac{8+3i}{4+3i}\times \frac{4-3i}{4-3i} \\ =\frac{\left(8\right)\left(4\right)+\left(8\right)\left(-3i\right)+\left(3i\right)\left(4\right)+\left(3i\right)\left(-3i\right)}{{\left(4\right)}^2-{\left(3i\right)}^2}\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \\ =\frac{32-24i+12i-9i^2}{16-9i^2} \\ =\frac{32-12i-9\left(-1\right)}{16-9\left(-1\right)}\left[\because i^2=-1\right] \end{gathered}$$

Further simplify the above expression.

$$\begin{gathered} \frac{8+3i}{4+3i}=\frac{32-12i+9}{16+9} \\ =\frac{41-12i}{25} \\ =\frac{41}{25}-\frac{12i}{25} \\ =1.64-0.48i \end{gathered}$$

Thus, the simplified form of the given expression is $1.64-0.48i$.

Part b. Step 1. Definition of complex number.

The general form of a complex number is $z=a+bi$, where $i=\sqrt{-1}\Rightarrow i^2=-1$.

Part b. Step 2. Given Information.

The given expression is:

$\sqrt{-10}\cdot \sqrt{-40}$

Part b. Step 3  Simplify the given expression.

The given expression can be written as:

$$\begin{gathered} \sqrt{-10}\cdot \sqrt{-40}=\sqrt{-1\cdot 10}\cdot \sqrt{-1\cdot 40} \\ =\sqrt{-1}\sqrt{10}\cdot \sqrt{-1}\sqrt{40}\left[\because \sqrt{ab}=\sqrt{a}\sqrt{b}\right] \\ =\sqrt{10}i\cdot \sqrt{40}i\left[\because i=\sqrt{-1}\right] \\ =\sqrt{10}\sqrt{40}{i}^2 \\ =\sqrt{400}\left(-1\right)\left[\because i^2=-1\right] \\ =20\left(-1\right) \\ =-20 \end{gathered}$$

The complex form$a+bi$ of is $-20+0i$.

Thus, the simplified form of the given expression is width="61" height="20" role="math">−20+0i.