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

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College Algebra

1. Equations & Inequalities

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1:54 minutes

Problem 35

Textbook Question

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

Verified step by step guidance

Recognize that both the numerator and the denominator involve square roots of negative numbers, which means we are dealing with imaginary numbers.

Express the square roots in terms of imaginary numbers: \( \sqrt{-24} = \sqrt{24} \cdot i \) and \( \sqrt{8} = \sqrt{8} \cdot i \).

Simplify the square roots: \( \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \) and \( \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \).

Substitute the simplified forms back into the expression: \( \frac{2\sqrt{6} \cdot i}{2\sqrt{2} \cdot i} \).

Cancel out the common terms \( 2i \) from the numerator and the denominator, leaving \( \frac{\sqrt{6}}{\sqrt{2}} \), and simplify further by rationalizing the denominator if necessary.

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Key Concepts

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

Square Roots of Negative Numbers

In algebra, the square root of a negative number involves the imaginary unit 'i', where i = √-1. For example, √-24 can be simplified to √(24) * √(-1) = 2√6 * i. Understanding this concept is crucial for handling expressions that include square roots of negative values.

Simplifying Square Roots

Simplifying square roots involves breaking down the radicand (the number under the square root) into its prime factors and extracting perfect squares. For instance, √8 can be simplified to √(4 * 2) = 2√2. This process is essential for simplifying expressions in algebra.

Division of Complex Numbers

When dividing complex numbers, such as those involving imaginary units, it is important to express the result in a standard form. This often involves multiplying the numerator and denominator by the conjugate of the denominator. In this case, simplifying √-24 / √8 requires careful handling of both real and imaginary components.