Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. \\(\sqrt{-91}\\)

Explanation:

Step1: Recall the definition of imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-a} \) (where \( a>0 \)) as \( \sqrt{a}\cdot\sqrt{-1}=\sqrt{a}i \).
For \( \sqrt{-91} \), we can split the square root:
\( \sqrt{-91}=\sqrt{91\times(-1)} \)

Step2: Apply the property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0, b\geq0 \); here we extend it to complex numbers where \( b = - 1\)), we get:
\( \sqrt{91\times(-1)}=\sqrt{91}\cdot\sqrt{-1} \)
Since \( \sqrt{-1}=i \), we substitute that in:
\( \sqrt{91}\cdot i = i\sqrt{91} \)

Answer:

\( i\sqrt{91} \)