2. Equations and Inequalities

Multiply Complex Numbers

Complex Numbers

Complex Numbers: Multiplying and Dividing in Polar Form, Ex 1

Adding and Subtracting Complex (Imaginary) Numbers

In Exercises 95–99, perform the indicated operations and write the result in standard form.

(1 + i)/(1 + 2i) + (1 - i)/(1 - 2i)

Complex Numbers:  Multiplying - Ex 2

Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2

In Exercises 9–20, find each product and write the result in standard form.

(7 - 5i)(- 2 - 3i)

Find each product. Write answers in standard form. (2+i)²

Pre-Calculus - Simplify powers of i

Write each number as the product of a real number and i. -√-18

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) √24

Subtract complex numbers: combining like terms

Find each quotient. Write answers in standard form. 2 / 3i

In Exercises 37–52, perform the indicated operations and write the result in standard form.

√-64 - √-25

In Exercises 9–20, find each product and write the result in standard form.

(5 - 2i)^2

In Exercises 29–36, simplify and write the result in standard form.

√-108

In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(3√-5)(- 4√-12)

In Exercises 29–36, simplify and write the result in standard form.

√-49

In Exercises 1–8, add or subtract as indicated and write the result in standard form.

6 - (- 5 + 4i) - (- 13 - i)

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) -6 -2i

Find each product. Write answers in standard form. (-2-3i)(-2+3i)

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.

i^135

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) 0

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form.

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

Find each quotient. Write answers in standard form. 1-3i / 1+i

Complex Numbers: Dividing - Ex 2

Simplify positive integer powers of i

Find each product. Write answers in standard form. (2+i)(3-2i)

Decide whether each statement is true or false. If false, correct the right side of the equation. (-2+7i) - (10-6i)= -12+i

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(- 3 - √-7)^2

Decide whether each statement is true or false. If false, correct the right side of the equation. √-25 = 5i

Write each number as the product of a real number and i. √-288

Find each product. Write answers in standard form. i(3-4i)(3+4i)

In Exercises 61–64, write each complex number in standard form.

(1 - i)^3

In Exercises 9–20, find each product and write the result in standard form.

- 3i(7i - 5)

Complex Numbers: Operations, Complex Conjugates, and the Linear Factorization Theorem

In Exercises 95–99, perform the indicated operations and write the result in standard form.

(i^85 - i^83)/i^45

In Exercises 61–64, write each complex number in standard form.

(1 + i)^3

In Exercises 61–64, write each complex number in standard form.

(2 + 3i)^3

In Exercises 21–28, divide and express the result in standard form.

(2 + 3i)/(2 + i)

Find each product. Write answers in standard form. (3-i)(3+1)(2-6i)

In Exercises 21–28, divide and express the result in standard form.

5i/(2 - i)

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.

i^114

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) -7i

Add complex numbers by combining like terms

Find each quotient. Write answers in standard form. 8 / -i

Perform each operation. Write answers in standard form. -12 -i / -2 -5i

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.

i^44

Find each product. Write answers in standard form. (1+3i)(2-5i)

In Exercises 95–99, perform the indicated operations and write the result in standard form.

4/(2 + i)(3 - i)

In Exercises 29–36, simplify and write the result in standard form.

√(1^2 - 4 × 0.5 × 5)

In Exercises 21–28, divide and express the result in standard form.

2/(3 - i)

In Exercises 95–99, perform the indicated operations and write the result in standard form.

(i^98 - i^94)/i^49

Rewriting Powers of  ' i  '  - Ex 1

Find each product or quotient. Simplify the answers. √-10 / √-40

Find each quotient. Write answers in standard form. -5 / i

Perform each operation. Write answers in standard form. (5-11i)(5+11i)

Perform Operations with Square Roots of Negative Numbers

Find each product or quotient. Simplify the answers. √-6 * √-2 / √3

In Exercises 9–20, find each product and write the result in standard form.

(3 + 5i)(3 - 5i)

Operations with Square Roots with Negative Numbers in the Radicand

Write each number as the product of a real number and i. √-10

Perform each operation. Write answers in standard form. (6-i) + (7-2i)

Find each product. Write answers in standard form. (√6+i)(√6-i)

Work each problem. Show that -3+4i is a solution of the equation x²+6x+25=0.

Find each sum or difference. Write answers in standard form. 3√7 - (4√7-i) -4i + (-2√7+5i)

Write each number in standard form a+bi. -6-√-24 / 2

In Exercises 1–8, add or subtract as indicated and write the result in standard form.

(3 + 2i) - (5 - 7i)

Find each product or quotient. Simplify the answers. √-13 * √-13

In Exercises 48–57, perform the indicated operations and write the result in standard form. (7-i)(2+3i)

In Exercises 21–28, divide and express the result in standard form.

- 6i/(3 + 2i)

Complex Numbers:  Multiplying - Ex 1

Write each number in standard form a+bi. -3+ √-18 / 24

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form.

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

Find each sum or difference. Write answers in standard form. (-4-i) - (2+3i) + (-4+5i)

In Exercises 9–20, find each product and write the result in standard form.

(2 + 3i)^2

In Exercises 21–28, divide and express the result in standard form.

8i/(4 - 3i)

Find each product. Write answers in standard form. i(2+7i)(2-7i)

In Exercises 1–8, add or subtract as indicated and write the result in standard form.

(7 + 2i) + (1 - 4i)

Simplifying Negative Radicals

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form.

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

Find each product or quotient. Simplify the answers. √-3 * √-8

Multiplying square roots of negative numbers

Perform each operation. Write answers in standard form. (-8+2i)(-1+i)

Write each number as the product of a real number and i. √-25

Complex Numbers: Dividing - Ex 1

Find each product. Write answers in standard form. 3i(2-i)²

Multiply complex numbers

In Exercises 37–52, perform the indicated operations and write the result in standard form.

5√-16 + 3√-81

Multiply two complex numbers

In Exercises 9–20, find each product and write the result in standard form.

(- 5 + 4i)(3 + i)

Find each sum or difference. Write answers in standard form. (-2+4i) - (-4+4i)

Perform each operation. Write answers in standard form. 15i- (3+2i) -11

Work each problem. Show that √2/2 + √2/2 i is a square root of i.

Decide whether each statement is true or false. If false, correct the right side of the equation. i^12 = 1

Simplifying a power of i

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form.

5√-16 + 3√-81

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(- 2 + √-4)^2

Find each product. Write answers in standard form. (3+i)(3-i)

Rewriting Powers of  ' i  '  - Ex 2

Write each number in standard form a+bi. 10+ √-200 / 5

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form.

(8 + 9i)(2 - i) - (1 - i)(1 + i)

In Exercises 9–20, find each product and write the result in standard form.

(- 5 + i)(- 5 - i)

In Exercises 95–99, perform the indicated operations and write the result in standard form.

8/(1 + 2/i)

Complex Numbers: Dividing - Ex 3

Find each sum or difference. Write answers in standard form. (-3+2i) - (-4+2i)

Find each product or quotient. Simplify the answers. √-30 / √-10

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(- 6 - √-12)/48

Complex Numbers - Graphing, Adding, Subtracting

In Exercises 29–36, simplify and write the result in standard form.

√(3^2 - 4 × 2 × 5)

In Exercises 21–28, divide and express the result in standard form.

2i/(1 + i)

Imaginary roots of negative numbers | Imaginary and complex numbers

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.

i^31

In Exercises 48–57, perform the indicated operations and write the result in standard form. √ (-32) - √ (-18)

Find each sum or difference. Write answers in standard form. (3+2i) + (9+3i)

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) √-36

Rationalize Denominator with Imaginary Part in Complex Number

Find each quotient. Write answers in standard form. 2-i / 2+i

In Exercises 1–8, add or subtract as indicated and write the result in standard form.

8i - (14 - 9i)

Subtract complex numbers

Evaluate higher powers of i

Perform each operation. Write answers in standard form. -5i(3-i)²

Find each product or quotient. Simplify the answers. √-24 / √8

Complex Numbers - Multiplying and Dividing

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(- 8 + √-32)/24

In Exercises 37–52, perform the indicated operations and write the result in standard form.

√-8 (√-3 - √5)

Use imaginary numbers to express square roots of negative numbers

Find each sum or difference. Write answers in standard form. (2-5i) - (3+4i) - (-2+i)

Learn the toughest concepts covered in Algebra with step-by-step video tutorials and practice problems by world-class tutors