Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
-√(-100)

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite the square root of a negative number as the square root of the positive number times \(i\). For \(\sqrt{-100}\), we can split it as \(\sqrt{100\times(- 1)}\).

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 the case where \(b = - 1\) with the definition of \(i\)), we have \(\sqrt{100\times(-1)}=\sqrt{100}\times\sqrt{-1}\). Since \(\sqrt{100} = 10\) and \(\sqrt{-1}=i\), then \(\sqrt{-100}=10i\).

Step3: Apply the negative sign

The original expression is \(-\sqrt{-100}\), so substituting \(\sqrt{-100}=10i\) into it, we get \(- 10i\).

Answer:

\(-10i\)