Textbook Question
Write each number as the product of a real number and i. √-25
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1. Equations & Inequalities
Complex Numbers
0:59 minutesProblem 29a
Textbook Question
Find each product or quotient. Simplify the answers. √-13 * √-13
Verified step by step guidance1
Recognize that the expression involves the product of two square roots of negative numbers: \(\sqrt{-13} \times \sqrt{-13}\).
Recall that \(\sqrt{-a} = \sqrt{a} \times i\) where \(i\) is the imaginary unit with the property \(i^2 = -1\).
Rewrite each square root using the imaginary unit: \(\sqrt{-13} = \sqrt{13} \times i\).
Substitute back into the product: \((\sqrt{13} \times i) \times (\sqrt{13} \times i)\).
Use properties of multiplication to combine terms: \((\sqrt{13} \times \sqrt{13}) \times (i \times i) = 13 \times i^2\), then simplify using \(i^2 = -1\).
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59sKey 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, where i² = -1. This allows for the definition and manipulation of square roots of negative numbers, such as √-13, which can be expressed as i√13.
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Square Roots of Negative Numbers
Properties of Square Roots
The product of square roots can be simplified using the property √a * √b = √(a*b), provided a and b are non-negative. When dealing with negative numbers under the root, this property is applied carefully by factoring out the imaginary unit i.
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Simplifying Products of Imaginary Numbers
When multiplying imaginary numbers, use the fact that i² = -1 to simplify expressions. For example, (√-13)(√-13) equals (i√13)(i√13) = i² * 13 = -13, converting the product back into a real number.
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