Chapter 3, Problem 7T

a.

To determine

To determine the sum of complex numbers.

Answer to Problem 7T

The solution of $\left(3-2\text{i}\right)+\left(4+3\text{i}\right)$ is $7+\text{i}$ .

Explanation of Solution

Given information:

The complex number given is: $\left(3-2\text{i}\right)+\left(4+3\text{i}\right)$

Formula used:

The following formula is used:

$\left(\text{a}+\text{bi}\right)+\left(\text{c}+\text{di}\right)=\left(\text{a}+\text{c}\right)+\left(\text{b}+\text{d}\right)\text{i}$

Calculation:

The sum can be calculated by:

$\begin{array}{l}\left(3-2\text{i}\right)+\left(4+3\text{i}\right)\left[\left(\text{a}+\text{bi}\right)+\left(\text{c}+\text{di}\right)=\left(\text{a}+\text{c}\right)+\left(\text{b}+\text{d}\right)\text{i}\right]\\ =\left(3+4\right)+\left(-2+3\right)\text{i }\left[\text{Simplify}\right]\\ =7+\text{i}\end{array}$

Hence, the solution of $\left(3-2\text{i}\right)+\left(4+3\text{i}\right)$ is $7+\text{i}$ .

b.

To determine

To determine the difference of complex numbers.

Answer to Problem 7T

The solution of $\left(3-2\text{i}\right)-\left(4+3\text{i}\right)$ is $-1-5\text{i}$ .

Explanation of Solution

Given information:

The complex number is: $\left(3-2\text{i}\right)-\left(4+3\text{i}\right)$

Formula used:

The following formula is used:

$\left(\text{a}+\text{bi}\right)-\left(\text{c}+\text{di}\right)=\left(\text{a}-\text{c}\right)+\left(\text{b}-\text{d}\right)\text{i}$

Calculation:

The difference can be calculated by:

$\begin{array}{l}\left(3-2\text{i}\right)-\left(4+3\text{i}\right)\left[\left(\text{a}+\text{bi}\right)-\left(\text{c}+\text{di}\right)=\left(\text{a}-\text{c}\right)+\left(\text{b}-\text{d}\right)\text{i}\right]\\ =\left(3-4\right)+\left(-2-3\right)\text{i }\left[\text{Simplify}\right]\\ =-1-5\text{i}\end{array}$

Hence, the solution of $\left(3-2\text{i}\right)-\left(4+3\text{i}\right)$ is $-1-5\text{i}$ .

c.

To determine

To determine the product of complex numbers.

Answer to Problem 7T

The solution of $\left(3-2\text{i}\right)\left(4+3\text{i}\right)$ is $18+\text{i}$ .

Explanation of Solution

Given information:

The complex number given is: $\left(3-2\text{i}\right)\left(4+3\text{i}\right)$

Formula used:

The following formula is used:

$\left(\text{a}+\text{bi}\right)\left(\text{c}+\text{di}\right)=\left(\text{ac}-\text{bd}\right)+\left(\text{ad}+\text{bc}\right)\text{i}$

Calculation:

The product can be calculated by:

$\begin{array}{l}\left(3-2\text{i}\right)\left(4+3\text{i}\right)\left[\left(\text{a}+\text{bi}\right)\left(\text{c}+\text{di}\right)=\left(\text{ac}-\text{bd}\right)+\left(\text{ad}+\text{bc}\right)\text{i}\right]\\ =\left(12+6\right)+\left(9-8\right)\text{i }\left[\text{Simplify}\right]\\ =18+\text{i}\end{array}$

Hence, the solution of $\left(3-2\text{i}\right)\left(4+3\text{i}\right)$ is $18+\text{i}$ .

d.

To determine

To determine the solution of two complex numbers.

Answer to Problem 7T

The solution of $\frac{3-2\text{i}}{4+3\text{i}}$ is $\frac{6+17\text{i}}{25}$ .

Explanation of Solution

Given information:

The complex number given is: $\frac{3-2\text{i}}{4+3\text{i}}$

Formula used:

The following formula is used:

$\frac{\left(\text{a+bi}\right)}{\left(\text{c+di}\right)}=\frac{\left(\text{a+bi}\right)}{\left(\text{c+di}\right)}\cdot \frac{\left(\text{c}-\text{di}\right)}{\left(\text{c}-\text{di}\right)}=\frac{\left(\text{ac+bd}\right)\text{+}\left(\text{bc}-\text{ad}\right)\text{i}}{{\text{c}}^{\text{2}}{\text{+d}}^{\text{2}}}$

Calculation:

The solution can be calculated by:

$\begin{array}{l}\frac{3-2\text{i}}{4+3\text{i}}\left[\frac{\left(\text{a+bi}\right)}{\left(\text{c+di}\right)}=\frac{\left(\text{a+bi}\right)}{\left(\text{c+di}\right)}\cdot \frac{\left(\text{c}-\text{di}\right)}{\left(\text{c}-\text{di}\right)}=\frac{\left(\text{ac+bd}\right)\text{+}\left(\text{bc}-\text{ad}\right)\text{i}}{{\text{c}}^{\text{2}}{\text{+d}}^{\text{2}}}\right]\\ =\frac{\left(3-2\text{i}\right)\left(4-3\text{i}\right)}{\left(4+3\text{i}\right)\left(4-3\text{i}\right)}\left[\text{multiply the numerator and denominator by the complex conjugate of the denominator}\right]\\ =\frac{\left(12-6\right)+\left(9+8\right)\text{i}}{16+9}\\ =\frac{6+17\text{i}}{25}\end{array}$

Hence, the solution of $\frac{3-2\text{i}}{4+3\text{i}}$ is $\frac{6+17\text{i}}{25}$ .

e.

To determine

To determine the solution of a complex number

Answer to Problem 7T

The solution ${\text{i}}^{48}$ of is $1$

Explanation of Solution

Given information:

The complex number given is: ${\text{i}}^{48}$

Calculation:

$\begin{array}{l}{\text{i}}^{48}\\ ={\left(\sqrt{-1}\right)}^{48}\left[\text{i}=\sqrt{-1}\right]\\ ={\left[{\left(\sqrt{-1}\right)}^2\right]}^{24}\\ ={\left(-1\right)}^{24}\left[{\left(\sqrt{-1}\right)}^2=-1\right]\\ =1\left[\text{if the power is an even number, the negative sign changes to positive}\right]\end{array}$

Hence, the solution of ${\text{i}}^{48}$ is $1$

f.

To determine

To find the solution of complex numbers.

Answer to Problem 7T

The solution of $\left(\sqrt{2}-\sqrt{-2}\right)\left(\sqrt{8}+\sqrt{-2}\right)$ is $6-2\text{i}$ .

Explanation of Solution

Given information:

The complex number given is: $\left(\sqrt{2}-\sqrt{-2}\right)\left(\sqrt{8}+\sqrt{-2}\right)$

Formula used:

The following formula is used:

$\left(\text{a+bi}\right)\cdot \left(\text{c+di}\right)\text{=}\left(\text{ac}-\text{bd}\right)\text{+}\left(\text{ad+bc}\right)\text{i}$

Calculation:

The solution can be obtained by:

$\begin{array}{l}\left(\sqrt{2}-\sqrt{-2}\right)\left(\sqrt{8}+\sqrt{-2}\right)\\ =\left(\sqrt{2}-\text{i}\sqrt{2}\right)\left(\sqrt{8}+\text{i}\sqrt{2}\right)\left[\sqrt{-1}=\text{i}\right]\\ =\left(\sqrt{16}+2\right)+\left(2-\sqrt{16}\right)\text{i }\left[\left(\text{a+bi}\right)\cdot \left(\text{c+di}\right)\text{=}\left(\text{ac}-\text{bd}\right)\text{+}\left(\text{ad+bc}\right)\text{i}\right]\\ =\left(4+2\right)+\left(2-4\right)\text{i }\left[\text{simplify}\right]\\ =6-2\text{i}\end{array}$

Hence, the solution of $\left(\sqrt{2}-\sqrt{-2}\right)\left(\sqrt{8}+\sqrt{-2}\right)$ is $6-2\text{i}$ .