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Chapter 2: Problem 47

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$\sqrt{-20}$$

Short Answer

Expert verified
The complex conjugate of \(\sqrt{-20}\) is \(-2i\sqrt{5}\) and their multiplication yields \(20\).

Step by step solution

01

Expressing the given number in complex form

The given number to analyze is \(\sqrt{-20}\). This can be written in terms of i, where \(i\) is the imaginary unit with the property that \(i^2 = -1\). Thus, the number can be expressed as \(i\sqrt{20}\) or \(i * 2\sqrt{5}\).
02

Finding the complex conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. Therefore, the complex conjugate of \(2i\sqrt{5}\) is \(-2i\sqrt{5}\).
03

Multiplying the number by its conjugate

The product of a complex number and its conjugate is given by \((a+bi)(a-bi)\), which simplifies to \(a^2 + b^2\). For our complex number \(2i\sqrt{5}\) and its conjugate \(-2i\sqrt{5}\), their multiplication gives \((2i\sqrt{5})(-2i\sqrt{5})\), which simplifies to \(4*5\) or \(20\).

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