Question

express the radical using the imaginary unit, i. express your answer in simplified form. ±√(-49) = ±

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-49}\) as \(\sqrt{49\times(-1)}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), and here we extend it for the imaginary unit case), we have \(\sqrt{49\times(-1)}=\sqrt{49}\times\sqrt{-1}\).

Step3: Simplify the square roots

We know that \(\sqrt{49} = 7\) and \(\sqrt{-1}=i\), so \(\sqrt{49}\times\sqrt{-1}=7i\). Also, we have the \(\pm\) sign from the original expression, so \(\pm\sqrt{-49}=\pm7i\).

Answer:

\(7i\) (with the \(\pm\) sign considered as per the original problem, so the full answer is \(\pm 7i\))