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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0:59 minutes

Problem 29

Textbook Question

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

Verified step by step guidance

Recognize that the expression involves the product of two square roots: \( \sqrt{-13} \times \sqrt{-13} \).

Recall the property of square roots: \( \sqrt{a} \times \sqrt{a} = a \).

Apply this property to the expression: \( \sqrt{-13} \times \sqrt{-13} = -13 \).

Understand that the square root of a negative number involves imaginary numbers, but in this case, the product simplifies directly to a real number.

Conclude that the product of \( \sqrt{-13} \times \sqrt{-13} \) simplifies to \(-13\).

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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. This concept is crucial when dealing with square roots of negative numbers, as they cannot be expressed as real numbers. For example, √-1 is represented as 'i', and thus √-13 can be simplified to √13 * i.

Properties of Exponents

The properties of exponents govern how to manipulate expressions involving powers. When multiplying like bases, you add the exponents, and when squaring a product, you square each factor. In the case of √-13 * √-13, you can apply the property that states (√a)² = a, leading to the simplification of the expression.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing the expression to its simplest form, often by factoring out perfect squares or using properties of radicals. In this case, √-13 * √-13 simplifies to (√-13)², which equals -13, demonstrating how to handle and simplify products of square roots.