Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
-√(-47)

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite the square root of a negative number using this definition. For \(\sqrt{-a}\) (where \(a>0\)), we can write it as \(\sqrt{a\times(-1)}=\sqrt{a}\times\sqrt{-1}=\sqrt{a}i\).

Step2: Rewrite \(\sqrt{-47}\)

Applying the above rule to \(\sqrt{-47}\), we have \(\sqrt{-47}=\sqrt{47\times(-1)}=\sqrt{47}\times\sqrt{-1}=\sqrt{47}i\) (since \(47\) is a prime number, its square root cannot be simplified further).

Step3: Simplify the original expression

The original expression is \(-\sqrt{-47}\). Substituting the value of \(\sqrt{-47}\) from Step 2, we get \(-\sqrt{47}i\).

Answer:

\(-i\sqrt{47}\) (or \(-\sqrt{47}i\))