Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. -4 + \sqrt{-2}

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-2}\) as \(\sqrt{2\times(-1)}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0, b\geq0\), and in the complex number system we extend this to handle negative numbers), we have \(\sqrt{-2}=\sqrt{2\times(-1)}=\sqrt{2}\times\sqrt{-1}\).

Step3: Substitute \(i\) for \(\sqrt{-1}\)

Since \(\sqrt{-1} = i\), then \(\sqrt{-2}=\sqrt{2}i\).

Step4: Rewrite the original expression

The original expression is \(-4+\sqrt{-2}\), substituting \(\sqrt{-2}=\sqrt{2}i\) into it, we get \(-4 + \sqrt{2}i\).

Answer:

\(-4 + \sqrt{2}i\)