Problem 36 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ (-2+\sqrt{-11})^{2} $$

Step-by-Step Solution

Answer

The result of squaring this complex number is \(-7 - 4\sqrt{11}i\).

Step 1: Identify the Imaginary Unit

In the term \(\sqrt{-11}\), the negative sign under the square root indicates we are dealing with an imaginary number. Therefore, we can rewrite the term as \(\sqrt{11}i\) where \(i\) is the imaginary unit with property \(i^{2} = -1\). This makes the entire expression \((-2 + \sqrt{11}i)^{2}\)

Step 2: Square the Complex Number

Squaring a complex number requires us to square each term and twice the product of both terms. Applying the formula \((a + bi)^{2} = a^{2} + 2abi - b^{2}\), we get: \((-2)^{2} + 2(-2)*(\sqrt{11}i) - (\sqrt{11}i)^{2} = 4 -4\sqrt{11}i - 11\)

Step 3: Simplify the Expression

Combine the real and imaginary parts to get the expression in standard form: \(4 - 11 - 4\sqrt{11}i = -7 - 4\sqrt{11}i \).