Question

Perform the indicated operations, and write the result in the form

(a)$(3-2i)+(4+3i)$

(b)$(3-2i)-(4+3i)$

(c)$(3-2i)(4+3i)$

(d)$\frac{(3-2i)}{(4+3i)}$

(e)$i^{48}$

(f)$(\sqrt{2}-\sqrt{-2})(\sqrt{8}+\sqrt{-2})$

Short answer

(a) The result is$7+i$

(b) The result is$-1-5i$

(c) The result is$18+i$

(d) The result is$\frac{6}{25}-\frac{17}{25}i$

(e) The result is 1

(f) The result is$6-2i$

Step-by-step solution

Part a. Step 1. Definition of Complex numbers.

A complex number is an expression of the form$a+ib$ Where a and b are real numbers andwidth="54">i2=-1

Part a. Step 2. Concept of operations in complex numbers.

Addition

$(a+ib)+(c+id)=(a+c)+i(b+d)$

Subtraction

$(a+ib)-(c+id)=(a-c)+i(b-d)$

Multiplication

$(a+ib).(c+id)=(ac-bd)+i(ad+bc)$

Part a. Step 3. Solving the operations.

$$\begin{gathered} (3-2i)+(4+3i) \\ =(3+4)+(-2+3)i \\ =7+i \end{gathered}$$

Part b. Step 1. Definition of Complex numbers.

A complex number is an expression of the form$a+ib$ Where a and b are real numbers and$i^2=-1$

Part b. Step 2. Concept of operations in complex numbers.

Addition

$(a+ib)+(c+id)=(a+c)+i(b+d)$

Subtraction

$(a+ib)-(c+id)=(a-c)+i(b-d)$

Multiplication

$(a+ib).(c+id)=(ac-bd)+i(ad+bc)$

Part b. Step 3. Solving the operations.

$$\begin{gathered} (3-2i)-(4+3i) \\ =(3-4)+(-2-3)i \\ =-1-5i \end{gathered}$$

Part c. Step 1. Definition of Complex numbers.

A complex number is an expression of the form$a+ib$ Where a and b are real numbers and$i^2=-1$

Part c. Step 2. Concept of operations in complex numbers.

Addition

$(a+ib)+(c+id)=(a+c)+i(b+d)$

Subtraction

$(a+ib)-(c+id)=(a-c)+i(b-d)$

Multiplication

$(a+ib).(c+id)=(ac-bd)+i(ad+bc)$

Part c. Step 3. Solving the quadratic equation.

$$\begin{gathered} (3-2i)(4+3i) \\ =12+9i-8i+6 \\ =(12+6)+i(9-8) \\ =18+i \end{gathered}$$

Part d. Step 1. Definition of Complex numbers.

A complex number is an expression of the form$a+ib$ Where a and b are real numbers and$i^2=-1$

Part d. Step 2. Concept of division in Complex numbers.

To simplify the quotient $\frac{a+bi}{c+di}$, multiply the numerator and denominator by the complex conjugate of the denominator.

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

Part d. Step 3. Solving the operation.

$\frac{3-2i}{4+3i}$

The complex conjugate of$4+3i$is$\overline{4+3i}=4-3i$

Therefore,

$$\begin{gathered} \frac{3-2i}{4+3i}=\left(\frac{3-2i}{4+3i}\right)\left(\frac{4-3i}{4-3i}\right) \\ =\left(\frac{12-9i-8i-6}{16+9}\right) \\ =\frac{6-17i}{25} \\ =\frac{6}{25}-\frac{17}{25}i \end{gathered}$$

Part e. Step 1. Definition of Complex numbers.

A complex number is an expression of the form$a+ib$ Where a and b are real numbers andwidth="54">i2=-1

Part e. Step 2. Concept of operations in complex numbers.

Addition

$(a+ib)+(c+id)=(a+c)+i(b+d)$

Subtraction

$(a+ib)-(c+id)=(a-c)+i(b-d)$

Multiplication

$(a+ib).(c+id)=(ac-bd)+i(ad+bc)$

Part e. Step 3. Solving the operation.

$$\begin{gathered} i^{48}={\left(i^2\right)}^{24} \\ ={\left(-1\right)}^{24}(i^2=-1) \\ =1 \end{gathered}$$

Part f. Step 1. Definition of Complex numbers.

A complex number is an expression of the form$a+ib$ Where a and b are real numbers and$i^2=-1$

Part f. Step 2. Concept of operations in complex numbers.

Addition

$(a+ib)+(c+id)=(a+c)+i(b+d)$

Subtraction

$(a+ib)-(c+id)=(a-c)+i(b-d)$

Multiplication

$(a+ib).(c+id)=(ac-bd)+i(ad+bc)$

Part f. Step 3. Solving the operation.

$$\begin{gathered} (\sqrt{2}-\sqrt{-2})(\sqrt{8}+\sqrt{-2}) \\ =(\sqrt{2}-\sqrt{2}i)(\sqrt{8}+\sqrt{2}i) \\ =\sqrt{2}.\sqrt{2}(1-i)(2+i) \\ =2(1-i)(2+i) \\ =2(2+i-2i+1) \\ =2(3-i) \\ =6-2i \end{gathered}$$