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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 30
Chapter 5, Problem 30

In Exercises 29–36, simplify and write the result in standard form.


√−196

Verified step by step guidance
1
Recognize that the expression involves the square root of a negative number, which means we are dealing with imaginary numbers in the complex number system.
Recall the definition of the imaginary unit: \(i = \sqrt{-1}\), so \(\sqrt{-a} = i\sqrt{a}\) for any positive real number \(a\).
Rewrite the expression \(\sqrt{-196}\) as \(\sqrt{-1 \times 196}\), which can be separated into \(\sqrt{-1} \times \sqrt{196}\).
Substitute \(\sqrt{-1}\) with \(i\) and simplify \(\sqrt{196}\) by finding its positive square root.
Express the final answer in standard form for complex numbers, which is $a + bi$, where \(a\) and \(b\) are real numbers.

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Key Concepts

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

Imaginary Numbers

Imaginary numbers extend the real number system by including the square root of negative one, denoted as i. This allows for the definition of the square root of negative numbers, such as √−196, which can be expressed as a multiple of i.
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Introduction to Complex Numbers

Simplifying Square Roots

Simplifying square roots involves factoring the radicand into perfect squares and other factors. For example, √196 is simplified to 14 because 196 is 14 squared. This process helps in rewriting expressions in their simplest form.
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Imaginary Roots with the Square Root Property

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit. Writing results in this form clearly separates the real and imaginary parts, which is essential for further operations and interpretations.
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Complex Numbers In Polar Form
Related Practice
Textbook Question

In Exercises 27–32, select the representations that do not change the location of the given point. (−2, 7π/6) (−2, −5π/6)

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Textbook Question

In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 7π/4 + i sin 7π/4)

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Textbook Question

In Exercises 29–36, simplify and write the result in standard form. ___ √−49

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Textbook Question

In Exercises 29–36, simplify and write the result in standard form. ____ √−108

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Textbook Question

In Exercises 27–32, select the representations that do not change the location of the given point. (2, − 3π/4) (2, − 7π/4)

Textbook Question

In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)