Complex Numbers and Operations Alıştırma Soruları — 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}

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Complex Division and Simplification

Tahmini Süre:1m 30s

Soru 2Soru →

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

Cevabı ve açıklamayı göster

Cevap: $8 - i$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

i^2 = -1

.

Tahmini Süre:1m 0s

Soru 3Soru →

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 $k$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 4Soru →

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

Cevabı ve açıklamayı göster

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

Cevap

\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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Simplifying complex expressions by reducing powers of

i

and rationalizing the denominator using the complex conjugate.

Soru 5Soru →

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

Cevabı ve açıklamayı göster

Cevap: 130

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Soru 6Soru →

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 $z$ ?

Cevabı ve açıklamayı göster

Cevap: -8

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

i

, and identification of the imaginary part.

Soru 7Soru →

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 $w$ ?

Cevabı ve açıklamayı göster

Cevap: 3.25

Cevap

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

.

Adım Adım Çözüm

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}

.

Anahtar Kavram

Absolute value of a complex number and operations on complex numbers

Tahmini Süre:2m 0s

Soru 8Soru →

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

Cevabı ve açıklamayı göster

Cevap: -6

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:45s

Soru 9Soru →

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}

Cevabı ve açıklamayı göster

Cevap: $-2$

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Division of complex numbers using the complex conjugate

Tahmini Süre:2m 0s

Soru 10Soru →

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 $z$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

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.

Adım Adım Çözüm

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

.

Anahtar Kavram

Division of complex numbers using the complex conjugate

Alternatif Yöntem

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.

Tahmini Süre:1m 30s

Soru 11Soru →

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

Cevabı ve açıklamayı göster

Cevap: $2 + i$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:45s

Soru 12Soru →

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 $k$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Rationalizing complex numbers and identifying real and imaginary components

Soru 13Soru →

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 $w$ ?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Soru 14Soru →

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

Cevabı ve açıklamayı göster

Cevap: 3 - 4i

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Simplifying complex numbers by evaluating powers of

i

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

Tahmini Süre:1m 30s

Soru 15Soru →

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

Cevabı ve açıklamayı göster

Cevap: $-5 + 15i$

Cevap

-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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

i^2 = -1

.

Tahmini Süre:1m 30s

Soru 16Soru →

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

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

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.

Adım Adım Çözüm

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

.

Anahtar Kavram

Expanding complex binomials and simplifying powers of the imaginary unit

Soru 17Soru →

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

i

-1

-i

1

0

Cevabı ve açıklamayı göster

Cevap: $-i$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Simplifying powers of the imaginary unit

i

Soru 18Soru →

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

Cevabı ve açıklamayı göster

Cevap: $1$

Cevap

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

.

Adım Adım Çözüm

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

).

Anahtar Kavram

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

Soru 19Soru →

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

Cevabı ve açıklamayı göster

Cevap: $24i$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Complex Numbers and Operations

Soru 20Soru →

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 $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Equality and multiplication of complex numbers

Complex Numbers and Operations

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm