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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1:48 minutes

Problem 33

Textbook Question

Find each product or quotient. Simplify the answers. √-30 / √-10

Verified step by step guidance

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

Express each square root in terms of imaginary numbers: \( \sqrt{-30} = \sqrt{30} \cdot i \) and \( \sqrt{-10} = \sqrt{10} \cdot i \), where \( i \) is the imaginary unit.

Rewrite the expression as a fraction of these imaginary numbers: \( \frac{\sqrt{30} \cdot i}{\sqrt{10} \cdot i} \).

Simplify the fraction by canceling out the \( i \) terms, resulting in \( \frac{\sqrt{30}}{\sqrt{10}} \).

Further simplify the expression by dividing the square roots: \( \sqrt{\frac{30}{10}} = \sqrt{3} \).

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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, typically expressed in the form a + bi, where 'i' is the imaginary unit defined as √-1. In the context of square roots of negative numbers, such as √-30 and √-10, we express these roots using 'i' to indicate their imaginary nature, allowing us to perform arithmetic operations on them.

Properties of Square Roots

The properties of square roots state that √(a/b) = √a / √b and √(a * b) = √a * √b. These properties are essential for simplifying expressions involving square roots, especially when dealing with products or quotients of square roots, as they allow us to break down complex expressions into simpler components.

Simplification of Expressions

Simplification of expressions involves reducing an expression to its simplest form, which often includes combining like terms, factoring, and rationalizing denominators. In the case of the expression √-30 / √-10, simplification will involve expressing the square roots in terms of complex numbers and then performing the division to arrive at a more manageable form.