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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1:20 minutes

Problem 21

Textbook Question

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

Verified step by step guidance

insert step 1> Start by recognizing that the square root of a negative number involves the imaginary unit i, where i^2 = -1.

insert step 2> Rewrite the expression \( \sqrt{-25} \) as \( \sqrt{25} \times \sqrt{-1} \).

insert step 3> Simplify \( \sqrt{25} \) to 5, since 25 is a perfect square.

insert step 4> Recognize that \( \sqrt{-1} \) is equal to i, the imaginary unit.

insert step 5> Combine the results to express the original expression as the product of a real number and i: 5i.

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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 do not have real solutions. For example, √-1 = i, and √-25 can be expressed as 5i, since √-25 = √(25) * √(-1).

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, typically expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. In the context of the question, √-25 can be represented as 0 + 5i, indicating that there is no real component and the entire value is imaginary.

Properties of Square Roots

The properties of square roots state that the square root of a product can be expressed as the product of the square roots of the individual factors. This property is crucial when dealing with negative numbers, as it allows us to separate the real and imaginary components. For instance, √-25 can be rewritten as √(25) * √(-1), leading to the conclusion that √-25 = 5i.