Textbook Question
In Exercises 9–20, find each product and write the result in standard form.(- 5 + 4i)(3 + i)
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1. Equations & Inequalities
The Imaginary Unit
2:20 minutesProblem 27
Textbook Question
Write each number as the product of a real number and i. -√-18
Verified step by step guidance1
Recognize that the expression involves a square root of a negative number, which means we will use the imaginary unit \(i\), where \(i = \sqrt{-1}\).
Rewrite \(-\sqrt{-18}\) as \(-\sqrt{-1 \times 18}\).
Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the terms: \(-\sqrt{-1} \times \sqrt{18}\).
Replace \(\sqrt{-1}\) with \(i\), giving us \(-i \times \sqrt{18}\).
Express \(\sqrt{18}\) as \(\sqrt{9 \times 2}\), which simplifies to \(3\sqrt{2}\), resulting in the final expression: \(-3i\sqrt{2}\).
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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 are defined as multiples of the imaginary unit 'i', where i is the square root of -1. They arise when taking the square root of negative numbers, which do not have real solutions. For example, √-1 = i, and thus √-18 can be expressed as √(18) * √(-1) = 3√2 * i.
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Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part, typically expressed in the form a + bi, where a is the real part and b is the coefficient of the imaginary part. In the context of the question, -√-18 can be rewritten as a complex number, specifically 0 - 3√2 * i, where the real part is 0.
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Properties of Square Roots
The properties of square roots state that √(a * b) = √a * √b, and this can be applied to negative numbers by separating the real and imaginary components. When dealing with negative values under a square root, it is essential to factor out -1 to express the result in terms of 'i'. This property is crucial for simplifying expressions involving square roots of negative numbers.
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