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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 49
Chapter 2, Problem 49

Perform the indicated operations and write the result in standard form. −8(−3−−5)\(\sqrt{-8}\) \(\left\)( \(\sqrt{-3}\) - \(\sqrt{-5}\) \(\right\))

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1
Recognize that the expression involves square roots of negative numbers, which means we will be working with imaginary numbers. Recall that \( \sqrt{-1} = i \).
Rewrite the square roots of negative numbers using \( i \): \( \sqrt{-8} = \sqrt{8} \cdot i \) and \( \sqrt{-3} = \sqrt{3} \cdot i \).
Simplify \( \sqrt{8} \) as \( 2\sqrt{2} \), so \( \sqrt{-8} = 2\sqrt{2}i \).
Substitute the expressions back into the original problem: \( (2\sqrt{2}i)(\sqrt{3}i - \sqrt{5}) \).
Distribute \( 2\sqrt{2}i \) across the terms inside the parentheses: \( 2\sqrt{2}i \cdot \sqrt{3}i - 2\sqrt{2}i \cdot \sqrt{5} \).

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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, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as √-1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as they allow us to extend the real number system.
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Square Roots of Negative Numbers

The square root of a negative number is not defined within the real number system, but it can be expressed using imaginary numbers. For example, √-8 can be simplified to 2√2i, where 'i' represents the imaginary unit. This concept is crucial for solving problems that involve square roots of negative values, as it allows for the manipulation of these expressions in algebraic operations.
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Standard Form of Complex Numbers

The standard form of a complex number is typically written as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as addition or multiplication, it is important to express the final result in this standard form for clarity and consistency. This involves combining like terms and ensuring that the imaginary unit 'i' is properly accounted for in the final expression.
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Related Practice
Textbook Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 1/(x - 1) + 5 = 11/(x - 1)

Textbook Question

Write each English sentence as an equation in two variables. Then graph the equation. The y-value is three decreased by the square of the x-value.

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Textbook Question

In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? S = C/(1 - r) for r

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Textbook Question

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 5(x - 2) - 3(x + 4) ≥ 2x - 20

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. x2−2x=2x^2 - 2x = 2

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. x2+4x=12 x^2 + 4x = 12

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