Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \sqrt{-64}

Explanation:

Step1: Rewrite the radicand

Recall that for a negative number under the square root, we can use the imaginary unit \(i = \sqrt{-1}\). So, we rewrite \(\sqrt{-64}\) as \(\sqrt{64\times(-1)}\).
\(\sqrt{-64}=\sqrt{64\times(-1)}\)

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 64\) and \(b=- 1\)), we get \(\sqrt{64}\times\sqrt{-1}\).
We know that \(\sqrt{64} = 8\) and \(\sqrt{-1}=i\).
So, \(\sqrt{64}\times\sqrt{-1}=8\times i = 8i\)

Answer:

\(8i\)