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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 47
Chapter 2, Problem 47

Perform the indicated operations and write the result in standard form. −6−−1248\(\frac{-6 - \sqrt{-12}\)}{48}

Verified step by step guidance
1
Identify the expression to simplify: \(\frac{-6 - \sqrt{-12}}{48}\).
Recognize that the square root of a negative number involves imaginary numbers. Rewrite \(\sqrt{-12}\) as \(\sqrt{12} \times \sqrt{-1}\), which is \(\sqrt{12}i\).
Simplify \(\sqrt{12}\) by factoring it into \(\sqrt{4 \times 3}\), which equals \(2\sqrt{3}\). So, \(\sqrt{-12} = 2\sqrt{3}i\).
Substitute back into the original expression: \(\frac{-6 - 2\sqrt{3}i}{48}\).
Separate the real and imaginary parts by dividing both terms in the numerator by 48: \(\frac{-6}{48} - \frac{2\sqrt{3}i}{48}\). Then simplify each fraction to write the expression in standard form \(a + bi\).

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Key Concepts

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

Complex Numbers and Imaginary Unit

Complex numbers include a real part and an imaginary part, where the imaginary unit 'i' is defined as √-1. Understanding how to express square roots of negative numbers using 'i' is essential for simplifying expressions like √-12.
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Simplifying Radicals

Simplifying radicals involves factoring the number inside the square root to extract perfect squares. For example, √12 can be simplified to 2√3, which helps in rewriting expressions in a simpler form before performing operations.
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Adding & Subtracting Unlike Radicals by Simplifying

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. Writing results in this form makes it easier to interpret and use complex numbers in further calculations.
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Related Practice
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. 4(3x - 2) - 3x < 3(1 + 3x) - 7

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Textbook Question

In Exercises 45–47, solve each formula for the specified variable. T = (A-P)/Pr for P

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Textbook Question

Use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.

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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. (x - 2)/2x + 1 = (x + 1)/x

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. x2+6x=7x^2 + 6x = 7

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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 = P + Prt for r

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