Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

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1:17 minutes

Problem 65

Textbook Question

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

Verified step by step guidance

Step 1: Recognize that the problem is asking you to multiply two complex numbers. A complex number is in the form of a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i^2 = -1.

Step 2: Use the distributive property to multiply the two complex numbers. This means you multiply each term in the first complex number by each term in the second complex number. So, you would multiply (√6 by √6), (√6 by -i), (i by √6), and (i by -i).

Step 3: Simplify the results of your multiplication. Remember that i^2 = -1, so whenever you see i^2, you can replace it with -1.

Step 4: Combine like terms. In this case, you will combine the real numbers (those without i) and the imaginary numbers (those with i).

Step 5: Write your final answer in standard form, which is a + bi. The real part (a) is the number without i, and the imaginary part (b) is the number with i.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Conjugates

The conjugate of a complex number a + bi is a - bi. When multiplying a complex number by its conjugate, the result is a real number, specifically a² + b². This property is useful for simplifying expressions and finding products, as it eliminates the imaginary part.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as multiplication, the result should be simplified to this form for clarity and consistency, making it easier to interpret the real and imaginary components.