Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \\(sqrt{-42}\\)

Explanation:

Step1: Recall the definition of imaginary unit

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-42}\) as \(\sqrt{-1\times42}\).

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 where \(a = - 1\) and \(b = 42\)), we have \(\sqrt{-1\times42}=\sqrt{-1}\times\sqrt{42}\).

Step3: Substitute \(i=\sqrt{-1}\)

Since \(\sqrt{-1}=i\), then \(\sqrt{-1}\times\sqrt{42}=i\sqrt{42}\).

Answer:

\(i\sqrt{42}\)