Question

1. \sqrt{-90}

Explanation:

Step1: Recall imaginary unit definition

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-90}\) as \(\sqrt{90\times(-1)}\).

Step2: Use square - root property

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), and we extend it to the case where \(b = - 1\) with the help of the imaginary unit), we have \(\sqrt{90\times(-1)}=\sqrt{90}\times\sqrt{-1}\).

Step3: Simplify \(\sqrt{90}\)

We factorize \(90 = 9\times10\), then \(\sqrt{90}=\sqrt{9\times10}\). Again, using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 9\geq0,b = 10\geq0\)), we get \(\sqrt{9\times10}=\sqrt{9}\times\sqrt{10}=3\sqrt{10}\).

Step4: Substitute back

Since \(\sqrt{-1}=i\), then \(\sqrt{90}\times\sqrt{-1}=3\sqrt{10}\times i = 3i\sqrt{10}\) (or \(3\sqrt{10}i\)).

Answer:

\(3\sqrt{10}i\) (or \(3i\sqrt{10}\))