Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 27a

Chapter 2, Problem 27a

Write each number as the product of a real number and i. -√-18

Verified step by step guidance

Recognize that the expression involves the square root of a negative number, which can be rewritten using the imaginary unit \(i\), where \(i = \sqrt{-1}\).

Rewrite the expression \(\sqrt{-18}\) as \(\sqrt{18} \times \sqrt{-1}\), which becomes \(\sqrt{18} \times i\).

Simplify \(\sqrt{18}\) by factoring it into \(\sqrt{9 \times 2}\), which equals \(\sqrt{9} \times \sqrt{2}\).

Calculate \(\sqrt{9}\) as 3, so \(\sqrt{18}\) simplifies to \(3\sqrt{2}\).

Combine the results to express the original number as the product of a real number and \(i\): \(3\sqrt{2} \times i\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Imaginary Unit (i)

The imaginary unit i is defined as the square root of -1, i.e., i² = -1. It allows us to express the square roots of negative numbers in terms of real numbers multiplied by i, enabling the extension of the real number system to complex numbers.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, separate it into the square root of the negative sign and the square root of the positive part. For example, √(-18) = √(-1) × √(18) = i × √(18), which can then be simplified further by factoring.

Prime Factorization and Simplification of Radicals

Simplifying radicals involves breaking down the number inside the square root into prime factors and extracting perfect squares. For √18, since 18 = 9 × 2, √18 = √9 × √2 = 3√2, which helps express the number in simplest radical form.

Recommended video:

Guided course

05:20

Expanding Radicals

Related Practice

Textbook Question

Solve each inequality. Give the solution set in interval notation. | 3x - 4 | < 2

Textbook Question

Solve each equation using the square root property. 27 - x² = 0

views

Textbook Question

Solve each equation. 0.5x+ (4/3)x= x+10

Textbook Question

Manufacturing to Specifications. A manufacturing firm wants to package its product in a cylindrical container 3 ft high with surface area 8π ft². What should the radius of the circular top and bottom of the container be? (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.)

Textbook Question

Solve each equation using the square root property. 48 - x² = 0

Textbook Question

Perform each operation. Write answers in standard form. -5i(3-i)²