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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0:36 minutes

Problem 31

Textbook Question

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

Verified step by step guidance

Recognize that both \( \sqrt{-3} \) and \( \sqrt{-8} \) involve square roots of negative numbers, which means they are complex numbers.

Express each square root in terms of imaginary numbers: \( \sqrt{-3} = \sqrt{3}i \) and \( \sqrt{-8} = \sqrt{8}i \), where \( i \) is the imaginary unit defined as \( i = \sqrt{-1} \).

Multiply the two expressions: \( (\sqrt{3}i) \times (\sqrt{8}i) \).

Use the property of multiplication for square roots: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). Therefore, \( \sqrt{3} \times \sqrt{8} = \sqrt{24} \).

Combine the imaginary units: \( i \times i = i^2 = -1 \). Therefore, the product becomes \( \sqrt{24} \times (-1) \).

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Key Concepts

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

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit 'i', where i is the square root of -1. They arise when taking the square root of negative numbers, which is not possible within the realm of real numbers. For example, √-3 can be expressed as i√3, allowing for operations involving negative square roots.

Properties of Square Roots

The properties of square roots include the product and quotient rules, which state that √a * √b = √(a*b) and √a / √b = √(a/b) for non-negative a and b. These properties can be extended to include imaginary numbers, enabling the simplification of expressions involving square roots of negative values.

Simplification of Expressions

Simplification of expressions involves reducing complex expressions to their simplest form. This includes combining like terms, factoring, and applying mathematical properties. In the context of the given problem, simplifying the product of two imaginary numbers requires careful application of the properties of square roots and the definition of imaginary numbers.