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Chapter 1: Problem 33

Perform the indicated operations and write the result in standard form. $$ (-2+\sqrt{-4})^{2} $$

Short Answer

Expert verified
The result of the operation in standard form is \(8 - 8i\).

Step by step solution

01

Rewrite the square root of a negative number in terms of \(i\)

The square root of \(-4\) is the same as the square root of \(4\) times the square root of \(-1\). The square root of \(4\) is \(2\), and the square root of \(-1\) is denoted as \(i\). So, rewrite \(\sqrt{-4}\) as \(2i\). This gives us \[(-2+2i)^{2}\].
02

Perform the squaring operation

Expanding the square, we use the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a=-2\) and \(b=2i\). This gives us \[(-2)^2 + 2*(-2)*(2i) + (2i)^2 = 4 - 8i -4i^2.\]
03

Simplify the expression

Remember that \(i^2\) is equal to \(-1\). Thus, the expression simplifies as \[4 - 8i +4 = 8 - 8i.\]

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