Complex Numbers and Operations Practice Questions — ACT

For the complex number $z = \frac{3 + 5i}{2 - i}$ , where $i = \sqrt{-1}$ , what is the real part of $z$ ?

\frac{1}{5}

\frac{11}{5}

\frac{3}{2}

\frac{1}{3}

-\frac{1}{5}

Show answer & explanation

Answer: $\frac{1}{5}$

Answer

The real part of the complex number is

\frac{1}{5}

.

To find the real part of the complex fraction, we multiply both the numerator and the denominator by the complex conjugate of the denominator,

2 + i

. This yields a denominator of

(2-i)(2+i) = 4 - i^2 = 5

. The numerator becomes

(3+5i)(2+i) = 6 + 3i + 10i + 5i^2 = 1 + 13i

. Writing the result in standard form gives

\frac{1}{5} + \frac{13}{5}i

, so the real part is

\frac{1}{5}

.

Step-by-Step Solution

1

Identify the complex conjugate of the denominator, which is

2 + i

, and set up the multiplication for both the numerator and the denominator of

z

.

\frac{(3 + 5i)(2 + i)}{(2 - i)(2 + i)}

Multiplying the numerator and denominator by the conjugate of the denominator allows us to simplify the expression and eliminate the imaginary unit from the denominator.

2

Expand the numerator using binomial multiplication and simplify using the identity

i^2 = -1

.

6 + 3i + 10i + 5i^2 = 6 + 13i - 5 = 1 + 13i

This determines the simplified complex expression in the numerator.

3

Expand the denominator and simplify using the identity

i^2 = -1

.

(2 - i)(2 + i) = 4 - i^2 = 4 - (-1) = 5

This simplifies the denominator into a real number.

4

Combine the simplified numerator and denominator to write

z

in standard form

a + bi

and identify the real part

a

.

z = \frac{1 + 13i}{5} = \frac{1}{5} + \frac{13}{5}i

, so the real part is

\frac{1}{5}

.

Expressing the complex number in standard form separates the real and imaginary components.

Key Concept

Complex Division and Simplification

Estimated Time:1m 30s

Question 2Question →

For the imaginary unit $i$ , what is the simplified form of the expression $(2 + 3i)(1 - 2i)$ ?

-4 - i

8 + i

8 - i

-4 + i

8 - 7i

Show answer & explanation

Answer: $8 - i$

Answer

8 - i

The correct answer is

8 - i

. To simplify the product of

(2 + 3i)(1 - 2i)

, we distribute the terms to get

2(1) + 2(-2i) + 3i(1) + 3i(-2i) = 2 - 4i + 3i - 6i^2

. Since

i^2 = -1

, the last term

-6i^2

simplifies to

-6(-1) = 6

. Combining the real parts (

2 + 6 = 8

) and the imaginary parts (

-4i + 3i = -i

) yields

8 - i

.

Step-by-Step Solution

1

Expand the product of the two binomials

(2 + 3i)(1 - 2i)

using the distributive property.

2 - 4i + 3i - 6i^2

Each term of the first binomial must be multiplied by each term of the second binomial.

2

Substitute

-1

for

i^2

in the expression.

2 - 4i + 3i - 6(-1) = 2 - 4i + 3i + 6

By definition, the imaginary unit

i

satisfies the equation

i^2 = -1

.

3

Combine the real terms and combine the imaginary terms to obtain the final simplified form.

8 - i

Adding the real parts (

2 + 6 = 8

) and the imaginary parts (

-4i + 3i = -i

) simplifies the expression to a single complex number.

Key Concept

Multiplying complex binomials and simplifying using the definition of the imaginary unit

i^2 = -1

.

Estimated Time:1m 0s

Question 3Question →

If the product of the complex numbers $4 - 2i$ and $k + 6i$ is a real number, where $k$ is a real constant and $i = \sqrt{-1}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

12

The product of the complex numbers

(4 - 2i)(k + 6i)

expands to

4k + 24i - 2ki - 12i^2

. Substituting

i^2 = -1

simplifies the expression to

(4k + 12) + (24 - 2k)i

. For this expression to represent a real number, the imaginary part must be zero:

24 - 2k = 0

, which yields

k = 12

.

Step-by-Step Solution

1

Multiply the two complex numbers

(4 - 2i)

and

(k + 6i)

using the FOIL method.

4k + 24i - 2ki - 12i^2

To find the product of the two complex expressions.

2

Substitute

i^2 = -1

into the expression and group the real and imaginary parts.

(4k + 12) + (24 - 2k)i

To simplify the expression into standard complex form

a + bi

.

3

Set the imaginary part of the resulting complex number to

0

and solve for

k

.

24 - 2k = 0 \implies k = 12

A complex number is real if and only if its imaginary part is equal to zero.

Key Concept

Complex multiplication and the definition of a real number in the complex plane

Question 4Question →

For the imaginary unit $i$ , which of the following is equivalent to the complex expression $\frac{2 + i^{15}}{1 - 2i^9}$ ?

\frac{4}{5} + \frac{3}{5}i

i

\frac{3}{5}i

\frac{4}{5}

-\frac{4}{3} - i

Show answer & explanation

Answer: $\frac{4}{5} + \frac{3}{5}i$

Answer

\frac{4}{5} + \frac{3}{5}i

Simplifying the powers of

i

yields

i^{15} = -i

and

i^9 = i

, giving the expression

\frac{2 - i}{1 - 2i}

. Multiplying the numerator and denominator by the complex conjugate

1 + 2i

results in the fraction

\frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

. Expanding both parts and substituting

i^2 = -1

gives

\frac{4 + 3i}{5}

, which simplifies to

\frac{4}{5} + \frac{3}{5}i

.

Step-by-Step Solution

1

Simplify the powers of the imaginary unit

i

in the expression.

i^{15} = i^{12} \cdot i^3 = 1 \cdot (-i) = -i

and

i^9 = i^8 \cdot i = 1 \cdot i = i

. The expression becomes

\frac{2 - i}{1 - 2i}

.

Reducing powers of

i

simplifies the expression and makes it easier to work with binomials.

2

Multiply the numerator and denominator by the complex conjugate of the denominator,

1 + 2i

.

\frac{2 - i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

Multiplying by the conjugate rationalizes the denominator, converting it to a real number.

3

Expand and simplify the numerator and denominator using the property

i^2 = -1

.

Numerator:

(2 - i)(1 + 2i) = 2 + 4i - i - 2i^2 = 2 + 3i - 2(-1) = 4 + 3i

. Denominator:

(1 - 2i)(1 + 2i) = 1 - 4i^2 = 1 - 4(-1) = 5

.

Expanding the binomial products allows combining real and imaginary parts.

4

Write the resulting fraction in standard complex form

a + bi

.

\frac{4 + 3i}{5} = \frac{4}{5} + \frac{3}{5}i

Standard form separates the real part and the imaginary part clearly.

Key Concept

Simplifying complex expressions by reducing powers of

i

and rationalizing the denominator using the complex conjugate.

Question 5Question →

Let $z$ be the complex number resulting from the product $(2 - 3i)(3 + i)$ , where $i = \sqrt{-1}$ . If $\bar{z}$ represents the complex conjugate of $z$ , what is the value of the product $z \cdot \bar{z}$ ?

32

58

81

95

130

Show answer & explanation

Answer: 130

Answer

130

The correct answer is obtained by first expanding

(2 - 3i)(3 + i)

using FOIL to get

6 + 2i - 9i - 3i^2

. Since

i^2 = -1

, this simplifies to

6 - 7i - 3(-1) = 9 - 7i

. The complex conjugate is

9 + 7i

. Multiplying these gives

9^2 - (7i)^2 = 81 - 49(-1) = 130

.

Step-by-Step Solution

1

Multiply the complex binomials to find

z

z = (2 - 3i)(3 + i) = 6 + 2i - 9i - 3i^2 = 6 - 7i - 3(-1) = 9 - 7i

To express the complex number in standard form

a + bi

, we expand the product using the distributive property and substitute

i^2 = -1

.

2

Determine the complex conjugate of

z

, denoted as

\bar{z}

\bar{z} = 9 + 7i

The complex conjugate of a complex number

a + bi

is

a - bi

, which is found by reversing the sign of the imaginary part.

3

Calculate the product of

z

and

\bar{z}

z \cdot \bar{z} = (9 - 7i)(9 + 7i) = 9^2 + 7^2 = 81 + 49 = 130

The product of a complex number

a + bi

and its conjugate

a - bi

is always a real number equal to

a^2 + b^2

.

Key Concept

Multiplying complex numbers and finding the product of a complex number and its conjugate.

Question 6Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = (3 - 2i)^2 - 4i^{103}$ . What is the imaginary part of $z$ ?

Show answer & explanation

Answer: -8

Answer

The imaginary part of

z

is

-8

.

The expression

(3-2i)^2

expands to

9 - 12i + 4i^2 = 5 - 12i

. The term

i^{103}

simplifies to

-i

since

103

leaves a remainder of

3

when divided by

4

. Subtracting

4i^{103}

corresponds to adding

4i

, giving

z = (5-12i) + 4i = 5-8i

. The coefficient of the imaginary part is

-8

.

Step-by-Step Solution

1

Expand

(3 - 2i)^2

5 - 12i

Use the binomial expansion formula and substitute

i^2 = -1

.

2

Simplify the term

-4i^{103}

4i

Since

103

divided by

4

leaves a remainder of

3

,

i^{103} = i^3 = -i

. Therefore,

-4i^{103} = -4(-i) = 4i

.

3

Combine terms to find

z

and identify its imaginary part

-8

Add the components:

z = (5 - 12i) + 4i = 5 - 8i

. The imaginary part is the coefficient of

i

, which is

-8

.

Key Concept

Operations on complex numbers including binomial expansion, powers of the imaginary unit

i

, and identification of the imaginary part.

Question 7Question →

For the imaginary unit $i = \sqrt{-1}$ , the complex number $w$ is defined as $w = \frac{5 + 12i}{(1 - i)^4}$ . What is the absolute value of $w$ ?

Show answer & explanation

Answer: 3.25

Answer

The absolute value of

w

is 3.25.

The correct answer is 3.25 because simplifying the denominator yields

(1-i)^4 = -4

. Dividing the numerator by

-4

gives the complex number

w = -1.25 - 3i

. The absolute value of

w

is then calculated as

\sqrt{(-1.25)^2 + (-3)^2} = \sqrt{1.5625 + 9} = \sqrt{10.5625} = 3.25

. Alternatively, using properties of absolute values, the absolute value of the quotient is the quotient of the absolute values:

|w| = \frac{|5 + 12i|}{|1-i|^4} = \frac{\sqrt{5^2 + 12^2}}{(\sqrt{1^2 + (-1)^2})^4} = \frac{13}{4} = 3.25

.

Step-by-Step Solution

1

Simplify the denominator

(1 - i)^4

(1 - i)^4 = -4

Calculate

(1 - i)^2 = -2i

, then square the result to obtain

(-2i)^2 = -4

.

2

Write the complex number

w

in standard form

a + bi

w = -1.25 - 3i

Divide each term in the numerator by the simplified denominator

-4

.

3

Calculate the magnitude

|w|

|w| = 3.25

Use the definition of absolute value of a complex number,

|a + bi| = \sqrt{a^2 + b^2}

.

Key Concept

Absolute value of a complex number and operations on complex numbers

Estimated Time:2m 0s

Question 8Question →

If $(5 - 2i) - (2 + ki) = 3 + 4i$ , where $i = \sqrt{-1}$ and $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: -6

Answer

The value of the constant

k

is

-6

.

Distributing the subtraction sign across the second complex number gives

(5 - 2i) - 2 - ki = 3 + 4i

. Combining the real parts (

5 - 2 = 3

) and grouping the imaginary parts yields

3 + (-2 - k)i = 3 + 4i

. Since the real parts are equal, we set the coefficients of the imaginary parts equal to each other:

-2 - k = 4

. Solving for

k

gives

k = -6

.

Step-by-Step Solution

1

Distribute the negative sign to the expression

(2 + ki)

-2 - ki

To remove the parentheses, the subtraction must apply to all terms inside the parentheses.

2

Group and combine the real parts and imaginary parts on the left side of the equation

3 - (2 + k)i

Grouping like terms allows us to express the left side as a standard complex number

a + bi

.

3

Equate the imaginary parts from both sides of the equation

-2 - k = 4

For two complex numbers to be equal, their corresponding real parts and imaginary parts must be equal.

4

Solve the linear equation for

k

k = -6

Isolating

k

gives the value that satisfies the original equation.

Key Concept

Equality of complex numbers and operations of addition/subtraction on complex numbers.

Estimated Time:45s

Question 9Question →

For the imaginary unit $i$ , if the complex number $z$ is defined by $z = \frac{(1 + 2i)^2}{2 - i}$ , what is the real part of $z$ ?

-2

-\frac{6}{5}

\frac{6}{5}

-\frac{10}{3}

-\frac{3}{2}

Show answer & explanation

Answer: $-2$

Answer

The real part of the complex number

z

is

-2

.

To find the real part of the complex number, we first simplify the expression by expanding the squared binomial in the numerator, which yields

-3 + 4i

. Next, we rationalize the fraction by multiplying both the numerator and the denominator by the complex conjugate of the denominator,

2 + i

. This multiplication yields

\frac{-10 + 5i}{5}

. Dividing both the real and imaginary terms by

5

results in the standard form

-2 + i

. Thus, the real part of this complex number is

-2

.

Step-by-Step Solution

1

Expand the squared binomial in the numerator of the expression for

z

.

(1 + 2i)^2 = 1^2 + 2(1)(2i) + (2i)^2 = 1 + 4i + 4i^2

Apply the algebraic identity

(a + b)^2 = a^2 + 2ab + b^2

.

2

Simplify the expanded numerator using the definition of the imaginary unit.

1 + 4i + 4(-1) = -3 + 4i

Since

i^2 = -1

, the term

4i^2

simplifies to

-4

. Combining this with

1

gives the real part

-3

.

3

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator to rationalize it.

z = \frac{-3 + 4i}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{(-3 + 4i)(2 + i)}{(2 - i)(2 + i)}

Multiplying the denominator by its complex conjugate,

2 + i

, eliminates the imaginary unit from the denominator.

4

Expand and simplify the numerator and denominator.

z = \frac{-6 - 3i + 8i + 4i^2}{4 - i^2} = \frac{-6 + 5i - 4}{4 - (-1)} = \frac{-10 + 5i}{5}

Using the distributive property in the numerator gives

-6 + 5i + 4i^2

. Since

i^2 = -1

, this simplifies to

-10 + 5i

. In the denominator,

(2-i)(2+i) = 4 - i^2 = 5

.

5

Divide both terms of the simplified numerator by the denominator to express

z

in standard form

a + bi

.

z = -2 + i

Dividing the real part

-10

by

5

yields the real part

-2

, and dividing the imaginary part

5i

by

5

yields the imaginary part

i

.

Key Concept

Division of complex numbers using the complex conjugate

Estimated Time:2m 0s

Question 10Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by $z = \frac{11 + 3i}{3 - i}$ . What is the real part of $z$ ?

Show answer & explanation

Answer: 3

Answer

The real part of the complex number is 3.

By multiplying both the numerator and the denominator of

\frac{11 + 3i}{3 - i}

by the conjugate of the denominator,

3 + i

, we obtain

\frac{(11+3i)(3+i)}{(3-i)(3+i)} = \frac{33 + 11i + 9i + 3i^2}{9 - i^2} = \frac{30 + 20i}{10} = 3 + 2i

. The real part of this complex number is the term without

i

, which is 3.

Step-by-Step Solution

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

3 + i

.

z = \frac{(11 + 3i)(3 + i)}{(3 - i)(3 + i)}

To eliminate the imaginary unit from the denominator.

2

Expand and simplify the numerator using the distributive property and substituting

-1

for

i^2

.

(11 + 3i)(3 + i) = 33 + 11i + 9i + 3i^2 = 33 + 20i + 3(-1) = 30 + 20i

To combine the real and imaginary terms of the numerator.

3

Expand and simplify the denominator using the difference of squares property and substituting

-1

for

i^2

.

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

To find the real number denominator.

4

Divide each term in the simplified numerator by the denominator.

z = \frac{30 + 20i}{10} = 3 + 2i

To express the complex number in the standard form

a + bi

.

5

Extract the real part of the resulting complex number

3 + 2i

.

3

The real part of a complex number in the form

a + bi

is

a

.

Key Concept

Division of complex numbers using the complex conjugate

Alternative Method

Instead of simplifying the fraction directly, assume the resulting complex number is

x + yi

, where

x

represents the real part and

y

represents the imaginary part. We can set up the equation

x + yi = \frac{11 + 3i}{3 - i}

and multiply both sides by

3 - i

to get

(x + yi)(3 - i) = 11 + 3i

. Expanding the left side gives

(3x + y) + (3y - x)i = 11 + 3i

. Equating the real and imaginary parts yields a system of linear equations:

3x + y = 11

and

-x + 3y = 3

. Multiplying the second equation by 3 and adding it to the first equation gives

10y = 20

, which means

y = 2

. Substituting

y = 2

back into the first equation yields

3x + 2 = 11

, which simplifies to

3x = 9

, or

x = 3

. The real part is therefore 3.

Estimated Time:1m 30s

Question 11Question →

For the imaginary unit $i$ , where $i^2 = -1$ , which of the following is equivalent to the expression $\frac{5}{2 - i}$ ?

2 - i

10 + 5i

2 + i

\frac{10 + 5i}{3}

2 + 5i

Show answer & explanation

Answer: $2 + i$

Answer

The simplified expression is

2 + i

.

To simplify the expression, multiply both the numerator and denominator by the complex conjugate of the denominator, which is

2 + i

. This results in

\frac{5(2+i)}{(2-i)(2+i)} = \frac{10+5i}{4-i^2}

. Since

i^2 = -1

, the denominator becomes

4 - (-1) = 5

. Dividing both terms in the numerator by

5

gives

2 + i

.

Step-by-Step Solution

1

Multiply the numerator and the denominator by the complex conjugate of the denominator,

2 + i

.

\frac{5(2 + i)}{(2 - i)(2 + i)}

Multiplying by the conjugate rationalizes the denominator, converting it into a real number.

2

Expand the numerator and the denominator, substituting

-1

for

i^2

.

\frac{10 + 5i}{4 - (-1)} = \frac{10 + 5i}{5}

Using the distributive property for the numerator and the difference of squares identity for the denominator, along with the definition

i^2 = -1

.

3

Divide each term in the numerator by the denominator.

2 + i

Distributing the division by

5

to both the real and imaginary parts of the numerator simplifies the expression to standard form.

Key Concept

Rationalizing the denominator of a complex fraction by multiplying by the complex conjugate of the denominator.

Estimated Time:45s

Question 12Question →

For the imaginary unit $i$ , where $i^2 = -1$ , let $z$ be the complex number defined by $z = \frac{10 + ki}{2 - i}$ , where $k$ is a real constant. If the imaginary part of $z$ is $4$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is

5

.

Multiplying the numerator and denominator of

z = \frac{10 + ki}{2 - i}

by the complex conjugate

2 + i

gives

z = \frac{(20 - k) + (10 + 2k)i}{5}

. The imaginary part is

\frac{10 + 2k}{5}

. Setting this expression equal to

4

and solving for

k

yields

k = 5

.

Step-by-Step Solution

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

2 + i

.

z = \frac{(10 + ki)(2 + i)}{(2 - i)(2 + i)}

To eliminate the imaginary unit from the denominator and express the complex number in standard form.

2

Expand both the numerator and the denominator, using the property

i^2 = -1

.

z = \frac{20 + 10i + 2ki + ki^2}{4 - i^2} = \frac{(20 - k) + (10 + 2k)i}{5}

To separate the real terms and imaginary terms in the numerator and simplify the denominator to a real number.

3

Express the complex number in standard form

a + bi

to identify the imaginary part.

z = \frac{20 - k}{5} + \left(\frac{10 + 2k}{5}\right)i

The imaginary part of a complex number is the coefficient of

i

, which is

\frac{10 + 2k}{5}

.

4

Set the imaginary part equal to

4

and solve the linear equation for

k

.

\frac{10 + 2k}{5} = 4 \implies 10 + 2k = 20 \implies 2k = 10 \implies k = 5

To find the specific value of the constant

k

that makes the imaginary part of

z

equal to

4

.

Key Concept

Rationalizing complex numbers and identifying real and imaginary components

Question 13Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $w$ is defined as $w = \frac{6 + 4i}{2i}$ . What is the imaginary part of $w$ ?

Show answer & explanation

Answer: -3

Answer

The imaginary part of

w

is

-3

.

Dividing each term in the numerator of

\frac{6 + 4i}{2i}

by the denominator

2i

yields

\frac{6}{2i} + \frac{4i}{2i}

, which simplifies to

\frac{3}{i} + 2

. Since

i^2 = -1

, the term

\frac{3}{i}

can be rationalized to

-3i

. Thus, the complex number in standard form is

2 - 3i

. The imaginary part is the real coefficient of

i

, which is

-3

.

Step-by-Step Solution

1

Divide each term in the numerator by the denominator.

w = \frac{6}{2i} + \frac{4i}{2i}

This separates the quotient into two simpler terms that can be simplified individually.

2

Simplify both terms.

w = \frac{3}{i} + 2

Reduce the fractions by dividing out common factors in both the numerators and the denominators.

3

Rationalize the denominator of the imaginary term.

\frac{3}{i} \cdot \frac{i}{i} = \frac{3i}{i^2} = \frac{3i}{-1} = -3i

Multiply the numerator and denominator by

i

to eliminate the imaginary unit from the denominator, using the property

i^2 = -1

.

4

Combine the real and imaginary parts into standard form

a + bi

.

w = 2 - 3i

Group the real constant and the simplified imaginary term together.

5

Identify the imaginary part of the complex number.

-3

The imaginary part of a complex number

a + bi

is the real coefficient

b

of the imaginary unit

i

.

Key Concept

Simplifying a quotient of complex numbers by dividing by a pure imaginary number.

Question 14Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by:

z = \frac{1 - 3i^{45}}{1 + i^{95}}

Which of the following is equivalent to $z^2$ ?

-3 + 4i

5 - 4i

3 - 4i

3

5

Show answer & explanation

Answer: 3 - 4i

Answer

3 - 4i

The correct answer is found by first simplifying the powers of

i

in the expression for

z

:

i^{45} = i

and

i^{95} = -i

, which yields

z = \frac{1 - 3i}{1 - i}

. Rationalizing the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator (

1 + i

) simplifies the expression to

z = 2 - i

. Finally, squaring this result using the binomial expansion formula gives

(2 - i)^2 = 4 - 4i + i^2 = 3 - 4i

.

Step-by-Step Solution

1

Simplify the high integer powers of the imaginary unit

i

by using the fact that powers of

i

repeat in a cycle of four:

i^1 = i

,

i^2 = -1

,

i^3 = -i

, and

i^4 = 1

.

i^{45} = i

and

i^{95} = -i

Since

45 = 4(11) + 1

, the remainder is

1

, so

i^{45} = i^1 = i

. Since

95 = 4(23) + 3

, the remainder is

3

, so

i^{95} = i^3 = -i

.

2

Substitute these simplified values back into the expression for

z

.

z = \frac{1 - 3i}{1 - i}

This sets up the fraction with simplified imaginary terms in both the numerator and the denominator.

3

Rationalize the denominator by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

1 + i

.

z = 2 - i

Multiplying by the conjugate eliminates the imaginary unit from the denominator:

\frac{(1 - 3i)(1 + i)}{(1 - i)(1 + i)} = \frac{1 + i - 3i - 3i^2}{1 - i^2} = \frac{4 - 2i}{2} = 2 - i

.

4

Calculate the value of

z^2

by squaring the simplified complex number

2 - i

.

z^2 = 3 - 4i

Using the binomial squaring formula

(a - b)^2 = a^2 - 2ab + b^2

, we expand

(2 - i)^2

to get

2^2 - 2(2)(i) + i^2 = 4 - 4i - 1 = 3 - 4i

.

Key Concept

Simplifying complex numbers by evaluating powers of

i

, rationalizing fractions with complex conjugates, and expanding complex binomials.

Estimated Time:1m 30s

Question 15Question →

Let $i$ be the imaginary unit such that $i^2 = -1$ . What is the simplified form of the expression $(1 + 2i)^2(3 - i)$ ?

-9 + 3i

-5 + 15i

19 + 7i

-13 + 15i

50i

Show answer & explanation

Answer: $-5 + 15i$

Answer

-5 + 15i

The correct answer is

-5 + 15i

. We first expand the squared binomial

(1 + 2i)^2

, which results in

1 + 4i + 4i^2

. Substituting

i^2 = -1

yields

-3 + 4i

. We then multiply this by the second binomial

(3 - i)

to get

-9 + 3i + 12i - 4i^2

. Substituting

i^2 = -1

one more time and combining like terms leads to the final simplified result of

-5 + 15i

.

Step-by-Step Solution

1

Expand the squared binomial

(1 + 2i)^2

1 + 4i + 4i^2

Before multiplying by the second binomial, we must apply the exponent to the first binomial according to the order of operations.

2

Substitute

i^2 = -1

to simplify the expression from Step 1

-3 + 4i

Since

i^2 = -1

, the term

4i^2

becomes

4(-1) = -4

, and combining the real parts gives

1 - 4 = -3

.

3

Multiply the simplified term by

(3 - i)

using the FOIL method

-9 + 3i + 12i - 4i^2

We distribute each term of the first binomial into the second binomial:

(-3)(3) = -9

,

(-3)(-i) = 3i

,

(4i)(3) = 12i

, and

(4i)(-i) = -4i^2

.

4

Simplify the resulting expression by combining like terms and substituting

i^2 = -1

-5 + 15i

Combining the imaginary parts gives

3i + 12i = 15i

. Substituting

i^2 = -1

into

-4i^2

gives

-4(-1) = +4

. Finally, combining the real parts yields

-9 + 4 = -5

.

Key Concept

Simplification of complex expressions involving binomial squaring and multiplication under the definition

i^2 = -1

.

Estimated Time:1m 30s

Question 16Question →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the real part of the complex number $z = (2 - i)^3 + 9i$ ?

Show answer & explanation

Answer: 2

Answer

The real part of the complex number is 2.

Expanding the expression

(2 - i)^3

yields

2 - 11i

. Adding

9i

gives

2 - 2i

. The real part of this complex number is the term without

i

, which is 2.

Step-by-Step Solution

1

Expand the squared binomial

(2 - i)^2

.

3 - 4i

Apply the identity

(a - b)^2 = a^2 - 2ab + b^2

and substitute

i^2 = -1

.

2

Multiply the result of the square by

(2 - i)

to calculate

(2 - i)^3

.

2 - 11i

Distribute the terms

(3 - 4i)(2 - i) = 6 - 3i - 8i + 4i^2

and substitute

i^2 = -1

.

3

Add

9i

to the simplified cube to find the complex number

z

.

2 - 2i

Combine the imaginary components:

-11i + 9i = -2i

.

4

Identify the real part of

z

.

2

The real part of a complex number

a + bi

is

a

.

Key Concept

Expanding complex binomials and simplifying powers of the imaginary unit

Question 17Question →

Given that $i = \sqrt{-1}$ , what is the value of the expression $i^{83}$ ?

i

-1

-i

1

0

Show answer & explanation

Answer: $-i$

Answer

The value of the expression is

-i

.

The expression

i^{83}

can be simplified by dividing the exponent 83 by 4. Because the remainder is 3, the expression is equivalent to

i^3

. Since

i^3 = i^2 \cdot i = (-1) \cdot i = -i

, the correct value is

-i

.

Step-by-Step Solution

1

Divide the exponent 83 by 4 to determine the remainder.

The quotient is 20 with a remainder of 3, which means

83 = 4 \times 20 + 3

.

Powers of the imaginary unit

i

repeat in a cycle of four:

i^1 = i

,

i^2 = -1

,

i^3 = -i

, and

i^4 = 1

.

2

Rewrite the expression using the rules of exponents and substitute the values of

i^4

and

i^3

.

i^{83} = i^{4(20) + 3} = (i^4)^{20} \cdot i^3 = (1)^{20} \cdot (-i) = -i

.

Since

i^4 = 1

, any power of

i

that is a multiple of 4 simplifies to 1, leaving only the remainder power to determine the final value.

Key Concept

Simplifying powers of the imaginary unit

i

Question 18Question →

For the complex number $z = \frac{4 + 2i}{3 - i}$ , where $i$ is the imaginary unit such that $i^2 = -1$ , what is the imaginary part of $z$ ?

\frac{5}{4}

\frac{6}{5}

\frac{1}{5}

1

-2

Show answer & explanation

Answer: $1$

Answer

The imaginary part of

z

is

1

.

To find the imaginary part of

z

, multiply the numerator and denominator of

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

by the complex conjugate of the denominator, which is

3 + i

. This simplifies the denominator to

9 - i^2 = 10

and the numerator to

12 + 10i + 2i^2 = 10 + 10i

. Dividing each term by

10

yields

1 + i

, which has an imaginary part of

1

.

Step-by-Step Solution

1

Identify the complex conjugate of the denominator.

The complex conjugate of the denominator,

3 - i

, is

3 + i

.

To divide complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator to make the denominator a real number.

2

Multiply the numerator and denominator of the fraction by the complex conjugate.

z = \frac{(4 + 2i)(3 + i)}{(3 - i)(3 + i)}

Multiplying by

\frac{3 + i}{3 + i}

is equivalent to multiplying by

1

, which preserves the value of the complex number.

3

Expand and simplify the numerator and denominator.

Numerator:

(4 + 2i)(3 + i) = 12 + 4i + 6i + 2i^2 = 12 + 10i + 2(-1) = 10 + 10i

. Denominator:

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

.

Apply the distributive property (FOIL method) and substitute

i^2 = -1

to simplify the expression.

4

Divide each term in the numerator by the denominator to find the imaginary part.

z = \frac{10 + 10i}{10} = 1 + i

, so the imaginary part is

1

.

Rewrite the fraction in standard form

a + bi

, where the imaginary part is

b

(the coefficient of

i

).

Key Concept

Simplifying a quotient of complex numbers by multiplying both the numerator and the denominator by the complex conjugate of the denominator.

Question 19Question →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the value of the expression $(3 + 2i)^2 - (3 - 2i)^2$ ?

0

10

24

24i

10 + 24i

Show answer & explanation

Answer: $24i$

Answer

The correct answer is

24i

.

The expression can be simplified by expanding each binomial term first. The first term,

(3 + 2i)^2

, expands to

9 + 12i + 4i^2

. Since

i^2 = -1

, this simplifies to

9 + 12i - 4 = 5 + 12i

. The second term,

(3 - 2i)^2

, expands to

9 - 12i + 4i^2

, which simplifies to

9 - 12i - 4 = 5 - 12i

. Subtracting the second simplified term from the first gives

(5 + 12i) - (5 - 12i) = 5 - 5 + 12i - (-12i) = 24i

.

Step-by-Step Solution

1

Expand the first squared binomial expression,

(3 + 2i)^2

.

(3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i

Apply the binomial squaring formula

(a+b)^2 = a^2 + 2ab + b^2

and use the property of the imaginary unit where

i^2 = -1

.

2

Expand the second squared binomial expression,

(3 - 2i)^2

.

(3 - 2i)^2 = 9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i

Apply the binomial squaring formula

(a-b)^2 = a^2 - 2ab + b^2

and use the property of the imaginary unit where

i^2 = -1

.

3

Subtract the second expanded expression from the first.

(5 + 12i) - (5 - 12i) = 5 + 12i - 5 + 12i = 24i

Distribute the negative sign to both terms of the second complex number and combine the real and imaginary parts.

Key Concept

Complex Numbers and Operations

Question 20Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{a + bi}{1 - 2i}$ , where $a$ and $b$ are real numbers. If $z = 4 + 3i$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is 5.

To find the value of

a + b

, we start with the equation

\frac{a + bi}{1 - 2i} = 4 + 3i

. Multiplying both sides by the denominator gives

a + bi = (4 + 3i)(1 - 2i)

. Expanding the right side using the distributive property, we get

a + bi = 4(1) + 4(-2i) + 3i(1) + 3i(-2i) = 4 - 8i + 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 - 5i - 6(-1) = 4 - 5i + 6 = 10 - 5i

. By comparing the real and imaginary parts of both sides, we find that

a = 10

and

b = -5

. The sum of these two values is

a + b = 10 + (-5) = 5

.

Step-by-Step Solution

1

Isolate the numerator by multiplying both sides by the denominator.

a + bi = (4 + 3i)(1 - 2i)

To solve for the variables

a

and

b

in the numerator, we clear the fraction by multiplying by the denominator.

2

Expand the product of the two complex numbers.

a + bi = 4 - 8i + 3i - 6i^2

Distribute each term of the first binomial to each term of the second binomial.

3

Simplify the expression using the definition of

i^2

.

a + bi = 10 - 5i

Since

i^2 = -1

, the term

-6i^2

becomes

+6

. Combine the real parts (

4 + 6 = 10

) and imaginary parts (

-8i + 3i = -5i

).

4

Equate the components and calculate

a + b

.

a = 10

,

b = -5

, and

a + b = 5

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Therefore,

a = 10

and

b = -5

. Summing these yields

10 + (-5) = 5

.

Key Concept

Equality and multiplication of complex numbers

Complex Numbers and Operations

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