Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \\(\sqrt{-49}\\)

Explanation:

Step1: Recall the definition of imaginary unit

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 \(a\geq0,b\geq0\), here we extend it to complex numbers), we have \(\sqrt{49\times(-1)}=\sqrt{49}\times\sqrt{-1}\).
Since \(\sqrt{49} = 7\) and \(\sqrt{-1}=i\), then \(\sqrt{49}\times\sqrt{-1}=7i\).

Answer:

\(7i\)