Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 53

Chapter 2, Problem 53

In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)

Verified step by step guidance

Step 1: Recognize that the denominator contains a complex number (5 + i). To simplify the expression, we need to eliminate the imaginary part from the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Step 2: The conjugate of (5 + i) is (5 - i). Multiply both the numerator and denominator of the fraction by (5 - i): $ \frac{6}{5+i} \cdot \frac{5-i}{5-i} $.

Step 3: Expand the numerator by distributing 6 across (5 - i): $ 6(5 - i) = 30 - 6i $.

Step 4: Expand the denominator using the difference of squares formula: $ (5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 25 + 1 = 26 $.

Step 5: Write the result as $ \frac{30 - 6i}{26} $. Simplify the fraction by dividing both the real and imaginary parts of the numerator by 26 to express the result 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

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, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations involving them, such as addition, subtraction, multiplication, and division.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. To express a complex number in standard form, it is often necessary to eliminate any imaginary numbers from the denominator of a fraction. This is typically achieved by multiplying the numerator and denominator by the conjugate of the denominator.

Conjugate of a Complex Number

The conjugate of a complex number a + bi is a - bi. The conjugate is useful in simplifying expressions involving complex numbers, particularly when dividing by a complex number. By multiplying by the conjugate, one can eliminate the imaginary part from the denominator, allowing the expression to be rewritten in standard form.

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Related Practice

Textbook Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. $x^{\frac{2}{5}} + x^{\frac{1}{5}} - 6 = 0$

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. 3/(2x - 2) + 1/2 = 2/(x - 1)

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 4 x + 1 = 0$

Textbook Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? 1/R = 1/R₁ + 1/R₂ for R₁

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Textbook Question

In Exercises 51–58, solve each compound inequality. - 3 ≤ x - 2 < 1

Textbook Question

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i³¹

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