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Chapter 2: Problem 68

Write the complex number in standard form. $$(\sqrt{-75})^{2}$$

Short Answer

Expert verified
The complex number \((\sqrt{-75})^2\) in standard form is -75.

Step by step solution

01

Identify the complex number

First, recognize that the \(\sqrt{-75}\) is a complex number because the square root of a negative number is not a real number. The negative sign under the square root can be expressed with the imaginary unit \(i\), where \(i^2 = -1\). We rewrite \(\sqrt{-75}\) as \(\sqrt{75}i\)
02

Simplify the square root

Now, we handle the square root of 75. The square root of 75 simplifies to 5\(\sqrt{3}\), as 75 = 25*3 and the \(\sqrt{25}=5\). So, our complex number becomes \(5\sqrt{3}i\)
03

Square the complex number

Now, we square the complex number, \((5\sqrt{3}i)^2\). This is equal to \(75i^2\)
04

Simplify the squared complex number

Now, we simplify \(75i^2\). Remember that \(i^2\) is equal to -1. So, our final answer becomes \(75*-1 = -75\)

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