Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 31

Chapter 5, Problem 31

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

Verified step by step guidance

Recognize that the expression involves the square root of a negative number, which means the result will be a complex number. Recall that $\sqrt{-a} = \sqrt{a} \times i$, where $i$ is the imaginary unit with the property $i^2 = -1$.

Rewrite the expression $\sqrt{-108}$ as $\sqrt{108} \times i$ to separate the imaginary unit from the real number under the root.

Simplify $\sqrt{108}$ by factoring 108 into its prime factors or perfect squares. For example, $108 = 36 \times 3$, and since $\sqrt{36} = 6$, you can write $\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}$.

Substitute back to get the expression in terms of $i$: $\sqrt{-108} = 6\sqrt{3} \times i$.

Write the final answer in standard form for complex numbers, which is $a + bi$. Since there is no real part here, the expression is $0 + 6\sqrt{3}i$.

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Key Concepts

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

Simplifying Square Roots of Negative Numbers

When simplifying the square root of a negative number, recognize that it involves imaginary numbers. The square root of a negative number can be expressed as the product of the imaginary unit 'i' (where i² = -1) and the square root of the corresponding positive number.

Prime Factorization for Simplifying Radicals

To simplify a square root, break down the number inside the root into its prime factors. Pair factors to extract them from under the root, simplifying the expression. For example, √108 can be factored into √(36 × 3) = 6√3.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the imaginary coefficient. After simplifying the radical, express the result in this form to clearly separate real and imaginary components.

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

√−196

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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 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ

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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)