Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
-√(-52)

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-a}\) (where \(a>0\)) as \(\sqrt{a}\cdot\sqrt{-1}=\sqrt{a}i\). For the expression \(-\sqrt{-52}\), we first rewrite the square root of the negative number.

Step2: Simplify the radical

We can factor \(52\) as \(4\times13\), so \(\sqrt{52}=\sqrt{4\times13}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)), we have \(\sqrt{4\times13}=\sqrt{4}\cdot\sqrt{13} = 2\sqrt{13}\). Now, going back to \(-\sqrt{-52}\), we use the definition of \(i\): \(-\sqrt{-52}=-\sqrt{52\times(-1)}=-\sqrt{52}\cdot\sqrt{-1}\). Substituting \(\sqrt{52} = 2\sqrt{13}\) and \(\sqrt{-1}=i\), we get \(- 2\sqrt{13}i\).

Answer:

\(-2\sqrt{13}i\)