Question

express the radical using the imaginary unit, i. express your answer in simplified form. ±√(-14) = ±

Explanation:

Step1: Recall the definition of \(i\)

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-a}\) (where \(a>0\)) as \(\sqrt{a}\cdot\sqrt{-1}\).
For \(\pm\sqrt{-14}\), we can split the square root: \(\pm\sqrt{-14}=\pm\sqrt{14\times(-1)}\)

Step2: Apply the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0, b\geq0\); here we extend it to complex numbers), we have \(\pm\sqrt{14\times(-1)}=\pm\sqrt{14}\cdot\sqrt{-1}\)
Since \(\sqrt{-1} = i\), this simplifies to \(\pm\sqrt{14}i\) or \(\pm i\sqrt{14}\)

Answer:

\(\pm i\sqrt{14}\)