Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \[(-2+2i)^{2}\].

Step 2: 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 \[(-2)^2 + 2*(-2)*(2i) + (2i)^2 = 4 - 8i -4i^2.\]

Step 3: Simplify the expression

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